THOUGHT AND KNOWLEDGE

The jury was facing a difficult decision in the case of People vs. Collins (cited in Arkes & Hammond, 1986). The robbery victim could not identify his assailant. All he could recall was that the robber was a woman with a blonde ponytail who, after the robbery, rode off in a yellow convertible driven by a Black man with a moustache and a beard. The suspect fit this description, but could the jury be certain “beyond a reasonable doubt” that the woman who was on trial was the robber? She was blonde and often wore her hair in a ponytail. Her codefendant “friend” was a Black man with a moustache, beard, and yellow convertible. If you were the attorney for the defense, you would stress the fact that the victim could not identify this woman as the robber. What strategy would you use if you were the attorney for the prosecution?

The prosecutor produced an expert in probability theory who testified that the probability of these conditions “co-occurring (being blonde plus having a ponytail plus having a Black male friend plus his owning a yellow convertible and so on, when these characteristics are independent) was 1 in 12 million. The expert testified that this combination of characteristics was so unusual that the jury could be certain “beyond a reasonable doubt” that she was the robber.

The jury returned a verdict of “guilty.”

Probabilistic Nature of the World

The theory of probabilities is nothing but common sense confirmed by calculation.

—Pierre Simon La Place (1951, p. 196)

The legal system recognizes that we can never have absolute certainty in legal matters. Instead, we operate with various degrees of uncertainty. In the United States, juries are instructed to decide that someone is guilty of a crime when they are certain “beyond a reasonable doubt.” This standard was adopted because there is always some small amount of doubt that the accused may be innocent. Jurors are instructed to operate under a different level of doubt when they are deciding about guilt or innocence in a civil case. In civil cases, they are told to deliver a verdict of guilty when the “preponderance of evidence” supports this decision. Thus, jurors are instructed to operate under two different levels of uncertainty depending on whether the case before them is criminal or civil. They need to be more certain when deciding that an accused party is guilty in a criminal case than in a civil case.

Probability is the study of likelihood and uncertainty. It plays a critical role in all professions and in most everyday decisions. All medical diagnoses and treatment decisions are inherently probabilistic, as are decisions made in business, college admissions, advertising, and research. Probability is the cornerstone of science; the laws of probability guide the interpretation of all research findings. Many of our leisure activities also rely on the principles of probability, most notably horse racing and card games. Every time you decide to take an umbrella, invest in the stock market, buy an insurance policy, or bet on a long shot in the Kentucky Derby, you are making a probability judgment. Other than the proverbial death and taxes, there is very little in life that is known with certainty. Because we live in a probabilistic world, critical thinking will require an understanding of probability.

There is good evidence that training in the use of probability will improve your ability to utilize probability values in an appropriate manner. In an investigation of the use of statistical thinking in everyday reasoning tasks, researchers concluded that, “This study indicated clearly that statistical training can enhance the use of statistical rules in reasoning about everyday life and can do so completely outside the context of training” (Fong, Krantz, & Nisbett, 1986, p. 280). In other words, although the thinking skills presented in this chapter will require the use of basic arithmetic and probably some concentrated effort, it is likely that you will be a better thinker for having worked through the problems.

Likelihood and Uncertainty

If your facts are wrong but your logic is perfect, then your conclusions are inevitably false. Therefore, by making mistakes in your logic, you have at least a random chance of coming to a correct conclusion.

—Christie-Davies’ Theorem (reference unknown, taken from a calendar)

If I flip a “fair” coin (i.e., one that is not biased, which means that either a head or tail is equally likely) into the air and ask you to guess the probability that it will land heads up, you would say that the probability of a head is 50% (or .50). This means that the coin is expected to land heads up half of the time. Although the word “probability” is used in several different ways, the definition of probability that is most useful in the present context is the number of ways a particular outcome (what we call a success) can occur divided by the number of possible outcomes (when each possible outcome is equally likely). It is a measure of how often we expect an event to occur in the long run. Success may seem like a strange word in this context, but you can think of it as the outcome in which you are interested. In this case, a success is getting the coin to land heads up. There is only one way for a coin to land heads ups, so the number of ways a success can occur in this example is one. What are all the possible outcomes of flipping a coin in the air? The coin can either land heads up or tails up. (I’ve never seen a coin land on its edge, nor have I ever seen a bird come along and carry it off while it’s flipped in the air, so I am not considering these as possible outcomes.) Thus, there are two possible outcomes, each of which is as likely to happen as the other. To calculate the probability of getting a coin to land heads up, compute the number of ways a head can occur (1), divided by the number of possible outcomes (2), or 1/2, an answer you already knew. Because many people find it easier to think in percentages than in fractions, 1/2 is sometimes changed to 50%.

Let’s try another example. How likely are you to roll a 5 in one roll of a die? As there is only one way for a 5 to occur, the numerator of the probability fraction is 1. A die is a 6-sided (cube) figure; thus there are 6 possible outcomes in one roll. If the die is not “loaded,” that is when each side of the die is equally likely to land facing up, the probability of rolling a 5 is 1/6, or approximately 17%.

What is the probability of rolling an even number in one roll of a fair die? To find this probability, consider the number of ways a success can occur. You could roll a 2, 4, or 6, all possible even numbers. Thus, there are 3 ways a success can occur out of 6 equally likely outcomes, so the probability of rolling an even number is 3/6 = 1/2.

What is the probability of rolling a whole number less than 7? If someone asked me to bet on this happening, I would put up my house, my children, and my meager savings account to make this bet. In other words, I would bet that this will happen. Let’s see why. The number of ways a whole number less than seven can occur in one roll of a die is 6 (1, 2, 3, 4, 5, or 6), and the number of possible outcomes is 6. Thus, the probability is 6/6, or 1. When a probability is equal to 1 (or 100%), it must happen; it is certain to occur.

What is the probability of rolling an 8 in one roll of a die? Again, I would put up everything I own, but this time I would bet against this occurrence. The number of ways an 8 can occur is 0. Thus, the probability of this occurring is 0; it cannot occur. This situation also reflects absolute certainty. Probabilities range from 0 (can never happen) to 1 (must happen). Probability values close to 0 or 1 represent events that are almost certain not to occur or almost certain to occur, while probabilities near .5 (50%) represent maximum uncertainty, because either outcome is equally likely, and thus there is no basis for predicting either one. This relationship is depicted in Figure 7.1.

Odds

It is often convenient to discuss probabilities in terms of odds. If a friend gives you 3-to-1 odds that his school’s championship tiddly-winks team will beat your school’s tiddly-winks team, this means that if 4 games were played, he would expect his team to win 3 of them. Authorities on organized sports (announcers, sports page editors, and almost everyone else) usually express their degree of belief in the outcome of a sporting event in terms of odds. (Betting odds like those posted at racetracks and boxing matches refer to the amount of money that has been bet on each contender and, thus, have a slightly different meaning from the one described here.) To convert odds to a probability, add the two numbers that are given (e.g., 3:1 = 4), use the first number as the numerator and the sum as the denominator (3/4), and this is the equivalent probability.

Figure 7.1 Probability and likehood.

The Laws of Chance

Uncertainty is an essential and integral facet of all forms of life and is, in principle, unavoidable.

—Gideon Keren and Karl Halvor Teigen (2001, p. 191)

The most important phrase in the last section was “in the long run.” Except for those special cases when the probability of an outcome is either 0% or 100%, we cannot know with certainty what will happen. I cannot know when I roll a die whether I will roll a five, but, if I keep rolling a fair die for many, many trials, I do know that about 17% of the time I will roll a five. I cannot know which roll will produce a five, but I do know approximately how many rolls will land with a 5 showing if I keep rolling for a long time. This is an important point. When we speak of the laws of chance (or laws of probability), we are referring to the ability to predict the number or percentage of trials on which a particular outcome will occur. With a large number of trials, I can be very accurate about the number of times a particular outcome will occur, but I cannot know which trials will yield a particular outcome. This means that I can make good “long run” predictions and poor “short run” predictions.

Let’s consider insurance as an applied example of this distinction. When you buy a life insurance policy (or any other type of insurance policy), you are making a bet with the insurance company. You agree to pay a certain amount of money to the insurance company each year. They agree to pay your beneficiary a certain amount of money when you die. There are many different types of life insurance policies available, but for the purposes of this chapter, we will consider the simplest. I will use some simple numbers to demonstrate the statistical point that I want to make—in real life, the actual costs and payoffs are different from those used in this example. Suppose that you are 30 years old and that you agree to pay the insurance company $1,000 per year. When you die, your beneficiary will receive $20,000. You are betting with the insurance company that you will die at a fairly young age (a bet that you hope to lose) so that you will have paid them only a small amount of money and your beneficiary will receive a larger amount of money. If you die before the age of 50, you win the bet. Ignoring complications like inflation and interest, you will have paid less than the $20,000 your beneficiary will receive if you die at a young age. The insurance company, on the other hand, will win the bet if you live to a ripe old age. If you die when you are 70 years old, you will have paid them $40,000 and your dearly beloved will receive only $20,000.

Insurance companies can make money because of the laws of chance. No one knows when you or anyone else will die, but insurance companies do know how many people aged 30 (the age at which you bought your policy) will die before they reach their 50th birthday. Thus, although no one can predict accurately the age at death for any single individual, we can use the laws of chance to predict how many people will live to be any particular age.

How We Explain Chance Events

At the beginning of the 20th century the father of modern science fiction, Herbert George Wells, said in his writings on politics, “If we want to have an educated citizenship in a modern technological society, we need to teach them three things: reading, writing, and statistical thinking.” At the beginning of the 21st century, how far have we gotten with this program? In our society, we teach most citizens reading and writing from the time they are children, but not statistical thinking.

—Gerd Gigerenzer (2003, para. 1)

Most people can understand the laws of chance when they think about tossing a coin or rolling a pair of dice, but find it more difficult to recognize how chance operates in complicated real-world settings. Unusual events—the sort that make you say “imagine that”—are often reported in the news. Consider, for example, this newspaper headline: “Three Sisters Give Birth on the Same Day” (Snell, Peterson, Albert, & Grinstead 2001, pp. 23–25). According to an article in the March 14, 2001 edition of the Point Stevens Journal, Karralee Morgan, Marrianne Asay, and Jennifer Hone (real people—real names—no names have been changed to protect anyone) are sisters who all gave birth on March 11, 1998 (as reported in Chance Magazine). It is hard to hear this story and NOT ask, “What is the probability that three sisters would give birth on the same day?” As it turns out, this is a tricky question because there are several ways of interpreting what is being asked. Consider these two possible interpretations:

What is the probability that these three sisters (Karralee, Marrianne, and Jennifer) will all give birth on March 11, 1998?

What is the probability that somewhere in the United States (or anywhere in the world) there will be three sisters (any set of three sisters) who all give birth on the same day (any day of any one year)?

Before I give you the answers, think about what is being asked and decide which of these two interpretations is more likely to occur. Did you understand that the situation described in Question #2 is much more likely than the situation described in Question #1? According to calculations presented in a magazine that teaches about chance events (Chance Magazine, Spring 2001), the answer for #1 is about 1 in 50 million, but the answer for #2 is about 1 in 6,000. It seems likely that most people are thinking about Question #2 when they ask about the chance of three sisters giving birth on the same day, but it is easy to see why these are difficult concepts because the question is prompted by situation #1. If we want to know about any three sisters on any day in any one year (there are 365 different possible days, so there are more ways for this to happen), it is much more likely than if we mean a specific three sisters on a specific day.

Degrees of Belief

Probability is sometimes used to express the strength of a belief about the likelihood of an outcome. This is a second definition for the term probability. For example, if you interview for a job and believe that the interview went well, you might assess the probability that you will be offered the job as 80%. This probability value was not mathematically derived by calculating the number of ways a success could occur divided by the number of possible outcomes. Instead, it indicates your degree of belief that the job will be offered. It suggests a moderate to high likelihood. If someone else interviewed for the same job and believed that his chance of being offered the job was 50%, it would be obvious that he was less confident about getting the job than you were.

It’s a Miracle (or Maybe Just Statistics)

Given enough opportunities, outlier anomalies—even seeming miracles—will occasionally happen.

—Michael Shermer (2008, para. 1)

It happens to all of us, admittedly not often, but it happens. Sometimes we are thinking of a friend, perhaps getting ready to call him when the phone rings, and it is he. Not exactly a miracle, but rare enough to make us think, “Now what is the probability of that happening?” But, even seeming miracles can be explained with probabilities. First, because of confirmation bias, the strong tendency to find evidence that confirms what we believe to be true and to ignore evidence that is contrary to our beliefs, we remember astonishing coincidences and not the vast array of every interactions in which we all engage. Shermer (2008) explained how we can explain something as weird as “death premonitions”—dreams in which someone we know has died or is about to die.

Shermer provides this rough analysis:

The average person has about five dreams a night, or 1,825 dreams a year. If we remember only a tenth of our dreams, then we recall 182.5 dreams a year. There are 300 million Americans, who thus produce 54.7 billion remembered dreams a year. Sociologists tell us that each of us knows about 150 people fairly well, thus producing a network grid of 45 billion personal relationship connections. With an annual death rate of 2.4 million Americans, it is inevitable that some of those 54.7 billion remembered dreams will be about some of these 2.4 million deaths among the 300 million Americans and their 45 billion connections. In fact, it would be a miracle if some death premonitions did not come true. (para. 4)

I have not checked Shermer’s math, but let’s assume that it is correct. Most of us cannot “get our heads” around numbers in the millions or billions—we are far better at marveling at our personal experiences and the anecdotes of others whom we know. A common theme in this chapter is that the laws of probability are not intuitive, and even though we may prefer to rely on personal experiences, we will make better decisions if we can think probabilistically.

Factors Affecting Judgments about Likelihood and Uncertainty

The odds against there being a bomb on a plane are a million to one, and against two bombs a million times a million to one. Next time you fly, cut the odds and take a bomb.

—Benny Hill (quoted in Bryne, 1988, p. 349)

There is a large body of research literature documenting the fact that most people are biased in their assessment of probabilities. We fail to appreciate the nature of randomness and have strong misconceptions about likelihood and uncertainty (Hill, 2012). This is not a surprising finding given that we can only use probabilities to understand events “in the long run,” and most of our everyday experiences are based on “short run” observations. For example, there is a large body of data that show that, on the average, people who smoke die at a younger age than those who do not (American Cancer Society, 2012). Most of us cannot discover this relationship on our own because we do not know the age of death for large numbers of people, but we may know one or two people who smoked two packs a day and lived into their 90s. This sort of personal experience would lead us to doubt the statistics that were collected from many people. Remember though, personal experience is not a good way to make many judgments about the world. Recall from the last chapter that experience is an expensive teacher, but not a good one.

The Search for Meaning

To live, it seems is to explain, to justify, and to find coherence among diverse outcomes, characteristics, and causes.

—Thomas Gilovich (1991, p. 22)

We seek causes for events that happen to us and to others, but most of us rarely consider the randomness of many events. We look for patterns and meaning, a quest that can often be helpful, but also can lead to beliefs that are groundless. For example, consider this true story: A student stopped in my office to talk with me. He told about an “amazing thing” that just happened to him. He was a student in a class of 15 students. Each student had to make an oral presentation, and the order in which they were to present was determined by drawing numbers from a box. “Guess who picked number 1?” he asked excitedly. I guessed that he did. “Exactly, and do you know the probability of that!” I did, it was 1/15 or approximately 7 percent. “Isn’t that amazing? Out of 15 people in the class, I picked number 1. How can you explain that?” I attributed this not-so-amazing outcome to chance; after all, someone had to pick number 1. He was certain that it was a sign of something; maybe the “gods” had intervened or his karma had gone berserk (whatever that means). He was looking for a cause that could explain this event, and he never considered just plain “chance.”

There seems to be a universal need to find meaning in the vast array of everyday events that confronts us, a trait that makes it difficult to recognize chance events. You may have heard about the controversy over the “Bible Code,” which is the title of a book that describes how letters sampled at regular intervals (for example, every 50^th letter) from the Bible had embedded in them the names of famous people and foretold events that would happen in the future (Drosnin, 1997). Statisticians howled with laughter (and scorn) at this book because chance alone would result in many words appearing in these letter strings. On the other hand, it is easy to understand how someone who does not understand concepts in probability might be persuaded by the seemingly amazing number of names and words that were embedded in long strings of letters.

The author of the Bible Code (Drosnin, 1997) challenged statisticians to use the same technique of letter sampling with “Moby Dick,” a book which presumably would not have embedded special messages about the future into its letter sequences. A group of Australian computer scientists (McKay, Bar-Natan, & Kalia, 1999) took up the challenge and found “M. L. King” near the phrase “to be killed by them,” “Kennedy” near the word “shoot,” and many more famous names and meaningful phrases (including “Princess Di” and “Lincoln”) in letter strings taken from Moby Dick, thus showing that mere chance would result in many meaningful names and phrases embedded in long strings of letters. In general, we tend to underestimate the importance of being able to think about probabilities. In an editorial on this topic, one columnist chided the American public for making fun of a politician’s poor grammar because we should be far more concerned about our politicians’ math skills, especially their poor understanding of probabilities (Cole, June 11, 2001). There are a many critical issues that depend on the ability to think with numbers—global warming, population growth, and the economy, to name a few.

Overconfidence

Most people … tend to overestimate the degree to which we are responsible for our own successes.

—Simon Gervais and Terrance Odean (2001, p. 1)

By definition, there is always some uncertainty in probabilistic events. Yet, research has shown that people tend to be more confident in their decisions about probabilistic events than they should be. Consider an example that Daniel Kahneman, a researcher in this area, likes to use. When he and his coauthors began working on a text on decision making, they were fairly confident that they would have it completed within a year, despite the fact that they knew that most books like the one they were writing take many years to complete. They believed that they would beat these “odds.” In fact, it took them several years to complete the text.

In an experimental investigation of the overconfidence phenomenon, people were asked to provide answers with a specified degree of confidence to factual questions (Kahneman & Tversky, 1979). Try it with this question: “I feel 98 percent certain that the number of nuclear plants operating in the world in 1980 was more than ______ and less than ______.” Fill in the blanks with numbers that reflect 98% confidence. The researchers investigating this effect found that nearly one-third of the time, the correct answer did not lie between the two values that reflected a 98% level of confidence. (The correct answer to this question is 189.) This result demonstrates that people are often highly confident when their high degree of confidence is unwarranted.

Have you ever bought a lottery ticket? Do you know what the odds are against your hitting the jackpot? The laws of probability dictate that you should expect to lose, yet countless numbers of people expect to win. In fact, a disturbing poll published in Money magazine revealed that almost as many people are planning for their retirement by buying lottery tickets (39%) as are investing in stocks (43%) (Wang, 1994).

Overconfidence about uncertain events is a problem even for experts in many fields where there is great uncertainty. In an analysis of political predictions (who is likely to win, for example), Silver (2011, para.3) wrote: “Experts have a poor understanding of uncertainty. Usually, this manifests itself in the form of overconfidence: experts underestimate the likelihood that their predictions might be wrong.” Overconfidence can be disastrous for financial investors. In a study of individual investors, two economists found that most people fail to recognize the role that chance plays in the stock market, so they tend to attribute gains to their own expertise in picking stocks and losses to external forces that they could not control. The result is that overconfident investors trade their stocks far too often because they believe that they are making wise choices (Gervais & Odean, 2001). If they could recognize the effects of random fluctuations in the stock market instead of attributing the changes to their own trading behaviors, they would have traded less often and ended up in better financial shape.

Using Probability

Without giving it much thought, we utilize probabilities many times each day. Let’s start with one of the few examples where probability values are made explicit. Many people begin each day by reading the weather forecast in the morning paper. What do you do when you read that the probability of rain is 80% today? Most people will head off to school or work toting an umbrella. What if it does not rain? Can we conclude that the forecaster was wrong? The forecast of an 80% probability of rain means that out of every 100 days when the weather conditions are like those on this particular day, there will be rain on 80 of them. Thus, a probability of rain is, like all probability values, based on what we would expect in the long run. Weather experts know that 80 out of 100 days will have rain, but the forecasters cannot know with absolute certainty which days it will rain.

Suppose that you are to be married on this hypothetical day, and a magnificent outdoor ceremony is planned. Suppose that an 80% probability of rain was forecast, and it did not rain. Would you believe that something other than chance was responsible for the good weather or that the absence of rain is a good (or bad) sign for your marriage? If you would interpret the good weather as a sign from the heavens or some other astral body, then you have demonstrated the point that was just made—we seek meaning in events, even events as seemingly uncontrollable as the weather, and we rarely consider plain old chance.

The number of instances in which we are given explicit probability values that have been computed for us is relatively small. One area where this practice is growing is in the use of medical information sheets that are designed to help patients understand the risks and benefits of taking a particular drug. The Food and Drug Administration requires that all oral contraceptive medications (birth control pills) be packaged with statistical information about the health risks associated with them. To arrive at an intelligent decision based on the information provided, potential oral contraceptive users must be able to understand the statistical summaries that are presented in the medical information sheets.

Consider the following: Suppose you read that the risk of developing heart disease is 10.5 times more likely for oral contraceptive users than for non-users. Most people will conclude from this information that taking oral contraceptives presents a substantial risk of heart disease. Suppose now that you are told that only 3.5 women out of 100,000 users will develop heart disease. You probably would interpret this sentence as meaning that there is little risk associated with oral contraceptive use. Consider the “flip side” of this information and think about how you would assess safety if you read that 99,996.5 women out of 100,000 users will not develop heart disease. Does it seem even safer? Another way of presenting the same information is to convert it to a percentage. There is only a .0035% chance that oral contraceptive users will develop heart disease. Most people would now consider the risk associated with oral contraceptive use to be minuscule.

Which of these statements is correct? They all are. The only way they differ is the way in which the statistical information is presented, and different ways of presenting the same statistical information lead to very different assessments of safety (Halpern, Blackman, & Salzman, 1989). It is important to keep this in mind when interpreting statistical information. There is a trend to provide consumers with statistical risk information so that they can make informed safety judgments about a diverse assortment of topics including how to treat a particular type of cancer and the safety of nuclear energy. Although the topic of risk is considered in more detail later in this chapter, keep in mind that the best way to convert risk probabilities to a meaningful value is to write out all of the mathematically equivalent values (i.e., X out of Y occurrences, number of times greater risk than a meaningful comparison event, number that will die, number that will not die). Graphic representations of relative risks can also be helpful when there are many values that need to be compared simultaneously. The use of spatial arrays is touted throughout this book (e.g., circle diagrams when interpreting syllogisms, graphic organizers to comprehend complex prose, tree diagrams for use in making sound decisions). One advantage that they confer in this situation is that they reduce the memory load in working memory and allow us to consider several different alternatives “at a glance.”

Games of Chance

People love to play games. From Las Vegas to Atlantic City, and in all of the small towns in between, and in many other countries in the world, people spend countless hours and dollars, euros, and pounds playing games of chance, skill, and semi-skill. For many people, the only serious consideration they have ever given to probability is when playing games of chance.

Cards. Card playing is a ubiquitous pastime, with small children playing “Fish” and “Old Maid,” while their older counterparts play Canasta, Bridge, Poker, Pinochle, Black Jack, Hearts, and too many others to mention. The uncertainty inherent in card games adds to the pleasure of playing (although the camaraderie and pretzels and beer also help).

Good card players, regardless of the game, understand and utilize the rules of probability. Let’s apply the definitional formula for probability to card games. For example, how likely are you to draw an Ace of Spades from a deck of 52 cards? The probability of this happening is 1/52, or approximately 2%, because there is only one Ace of Spades and 52 possible outcomes. How likely are you to draw an ace of any suit from a full deck of cards? If you have been following the probability discussion so far, you will realize that the answer is 4/52, or approximately 8%, because there are 4 aces in a 52-card deck.

Although some professional card players claim to have worked out careful plans that will help them to change the odds of winning in their favor, it is not possible to “beat the house” for most card games, no matter how good a player one is. It is always difficult to tell the extent to which these stories of successful gamblers are hype. Professional gamblers often enjoy bragging about their winnings and conveniently forget about the times when they lost. Furthermore, most of the self-proclaimed expert gamblers are selling their “winning system.” I hope that you recall from the chapters on reasoning and analyzing arguments that when an “expert” stands to gain from the sale of a product, the expert’s opinion becomes suspect.

According to Gunther (1977), Vera Nettick (who is a real person) is a lucky lady. While playing a game of bridge, she was dealt a hand that contained all 13 diamonds. Breathlessly, she won a grand slam with the once-in-a-lifetime card hand. Any statistician will be quick to point out that every possible combination of cards will be dealt to somebody sooner or later. Thus, Vera Nettick’s hand was no more unusual than any other card hand, although it certainly is more memorable. Can you guess how often such a hand would occur? Gunther (1977, p. 30) figured this out as follows:

There are roughly 635 billion possible bridge hands. Of these, eight might be called ‘perfect’ hands, though some are more perfect than others. To begin with, there are four perfect no-trump hands. Such a hand would contain all four aces, all four kings, all four queens, and one of the four jacks. Any of these four hands would be unequivocally perfect, because no bid could top it. Slightly less perfect, in descending order, are hands containing all the spades, all the hearts, all the diamonds, and all the clubs. If there are eight of these perfect hands in a possible 635 billion, the statistical probability is that such a hand will be dealt one in every 79 billion tries, give or take a few. Now all we have to do is estimate how many games of bridge are played every year and how many hands are dealt in each game. Using fairly conservative estimates, it turns out that a perfect hand should be dealt to some lucky bridge player, somewhere in the United States, roughly once every three or four years.

Consider the two card hands shown in Figure 7.2 on page 353. Every possible combination of cards is equally likely when the cards are dealt at random. This topic is also discussed in the following chapter on decision making.

Roulette

Roulette is often thought of as an aristocratic game. It is strange that it has gained this reputation, because it is a game of pure chance. Unlike most card games, there is no skillful way to play roulette. As you probably know, roulette is played by spinning a small ball inside a circular array of numbered and colored pockets. Eighteen of the pockets are red; eighteen are black; and two are green. Players can make a variety of bets. One possible bet is that the ball will land in a red pocket. What is the probability of this event when the ball is equally likely to land in any pocket? There are 18 red pockets out of 38 pockets (possible outcomes); therefore, the probability of the ball landing in a red pocket is 18/38. Because this is a number less than .5, we know that it will land in a red pocket slightly less than half of the time. Thus, if you kept betting on red, you would lose slightly more often than you would win. Suppose now you bet on black pockets. Again, the probability would be 18/38; and again, if you continue to bet on black pockets, you will lose more often than you win. Of course, sometimes you will win, and at other times you will lose, but after many spins—in the long run, you will lose at roulette.

Figure 7.2 Which of these two card hands are you more likely to be dealt from a wellshuffled deck of cards?

The odds or probability of winning at any casino game are always favorable to the “house,” otherwise casinos could not stay in business. Actually, there is one person who has been able to “beat” the roulette odds. One of my heroes is Al Hibbs, a scientist who gained fame for his work at the Jet Propulsion Laboratory in Pasadena, California, where much of the work on the U.S. space program is done. When he was a student, he used his knowledge of probability to run his original stake of $125 up to $6,300 at the Pioneer Club in Reno. Here is how he did it: Hibbs knows that although every number in a roulette wheel should be equally likely to occur, all manufactured devices have imperfections. These imperfections make some numbers more likely to occur than others. Hibbs and a friend recorded the results of 100,000 spins of roulette wheel to find the numbers that occurred most often. Accordingly, they bet on these numbers. Unfortunately, none of us can duplicate his success, because the wheels are now taken apart and reassembled with different parts each day. Thus, while each wheel is still imperfect, the imperfections differ from day to day.

Computing Probabilities in Multiple Outcome Situations

If the weather forecaster announced a 50% chance of rain on Saturday and a 50% chance of rain on Sunday, does that definitely mean that it will rain this weekend?

—Gigerenzer (2003, para. 2)

We are often concerned with the probability of two or more events occurring, such as getting two heads in two flips of a coin or rolling a six at least once in two rolls of a die. These sorts of situations are called multiple outcomes.

Using Tree Diagrams

Although it is relatively easy to understand that the probability of getting a head on one flip of a fair coin is 1/2, it is somewhat more difficult to know intuitively the probability of getting four heads in four flips of a fair coin. Although an example of flipping a coin may seem artificial, it is a good way of showing how probabilities combine over many trials. Let’s figure it out. (Follow along with me, even if you are math-phobic. The calculations and mathematical thinking are relatively easy, if you work along with the examples. Do not look at the next several figures and exclaim, “no way, I’ll just skip it.” It is important to be able to think with and about numbers.)

On the first flip, only one of two possible outcomes can occur; a head (H) or tail (T). What can happen if a coin is flipped twice? There are four possible outcomes: a head on the first flip and a head on the second (HH), a head on the first flip and a tail on the second (HT), a tail on the first flip and a head on the second (TH), and a tail on the first flip and a tail on the second (TT). Since there are four possible outcomes and only one way to get two heads, the probability of this event is 1/4 (again assuming that the coin is fair, that is, that getting a head is as likely as getting a tail). There is a general rule, the “and rule,” for calculating this value in any situation. When you want to find the probability of one event and another event (a head on the first and second flip), you multiply their separate probabilities. By applying the “and rule,” we find that the probability of obtaining two tails when a coin is flipped twice is equal to 1/2 × 1/2, which is 1/4. Intuitively, the probability of both events occurring should be less likely than either event alone, and it is.

A simple way to compute this probability is to represent all possible events with tree diagrams. Tree diagrams were used in the chapter on reasoning when we figured the validity of “if, then” statements. In this chapter, we will add probability values to the “branches” of the tree to determine the probability of different combinations of outcomes. Later in this book, I will return to tree diagrams once again as a way of generating creative solutions to problems.

On the first flip, either an H or T will land facing up. For a fair coin, the probability of a head is equal to the probability of a tail, which is equal to .5. Let’s depict this as follows:

When you flip a second time, either an H on the first flip will be followed by an H or T, or a T on the first flip will be followed by an H or T. The probability of a head or tail on the second flip is still .5. Outcomes from a second flip are added as “branches” on a tree diagram.

As you can see from this tree, there are four possible outcomes. You can use this tree to find out the probability of other events. What is the probability of getting exactly one H in two flips of a coin? Since there are two ways for this to occur, (HT) or (TH), the answer is 2/4 or 1/2. When you want to find the probability of two or more different outcomes, add the probability of each outcome. This is called the “or rule.” Another way to ask the same questions is, “What is the probability of getting either a head followed by a tail (1/4) or a tail followed by a head (1/4)?” The correct procedure is to add these values, which equals 1/2. Intuitively, the probability of two or more events occurring should be higher (more likely) than the probability of any one of them occurring, and it is.

We can only use the “or rule” and the “and rule” when the events we are interested in are independent. Two events are independent when the occurrence of one of them does not influence the occurrence of the other. In this example, what you get on the first flip of a coin does not influence what you get on the second flip. In addition, the “or rule” requires that the outcomes be mutually exclusive, which means that if one occurs, the other cannot occur. In this example, the outcomes are mutually exclusive because we cannot obtain both a head and a tail on any single flip. If you are still wondering about the answer to the question posed in the opening of this paragraph about whether it will rain this weekend given a 50% probability of rain on Saturday and a 50% probability of rain on Sunday, you should be able to “see” that if you replaced the word “heads” with “rain” and the word “tails” with “no rain” in the figures above, you have the same problem. The probability of rain on Saturday and Sunday would be 25% (.50 X .50). The probability of rain on at least one of these days is 75% (.25 + .25 + .25).

Let’s get away from the artificial example of coin flipping, and apply the same logic in a more useful context. I am sure that every student has, at some time or other, taken a multiple-choice test. (Some students like to call them multiple-guess tests.) Most of these tests have five alternative answers for each question. Only one of these is correct. Suppose also that the questions are so difficult that all you can do is guess randomly at the correct answer. What is the probability of guessing correctly on the first question? If you have no idea which alternative is the correct answer, then you are equally likely to choose any of the five alternatives, assuming that each alternative is equally likely to be correct. Because the sum of all possible alternatives must be 1.0, the probability of selecting each alternative, when they are equally likely, is .20. One alternative is correct, and four are incorrect, so the probability of selecting the correct alternative is .20. A tree diagram of this situation is shown here.

What is the probability of getting the first two multiple-choice questions correct by guessing? We will have to add a second branch to a tree that will soon be very crowded. To save space and to simplify the calculations, all of the incorrect alternatives can be represented by one branch labeled “incorrect.” The probability of being incorrect on any single question is .8:

The probability of two correct questions by guessing is .2 × .2, which equals .04. That means that this would happen by chance only 4% of the time. Suppose we extend our example to three questions. I will not draw a tree, but you should be able to see by now that the probability is .2 × .2 × .2 = .008. This is so unusual that it would occur less than 1% of the time by chance. What would you conclude about someone who got all three of these questions correct? Most people (professors are people, too) would conclude that the student was not guessing, and that she really knew something. Of course, it is possible that the student was just lucky, but it is so unlikely that we would conclude that something other than luck was responsible for the outcome we obtained. Let me point out a curious side to this kind of reasoning. Consider the plight of Sara. She took a fifteen question multiple-choice test in which every question had five alternatives. Sara got all fifteen questions wrong. Can you determine the probability of this happening by chance? I won’t draw the tree diagram to depict this situation, but it is easy to see that the probability of being wrong on one question is .80; therefore the probability of being wrong on all 15 questions is (.80)¹⁵. This is .80 times itself 15 times, which equals .0352. Because this would happen by chance only 3.52% of the time, can Sara present the argument to her professor that something other than chance determined this unusual result? Of course, Sara can make this argument, but would you be willing to believe her if you were her professor? Suppose she argued that she must have known the correct answer to every question. How else could she have avoided selecting it in all fifteen consecutive questions? I don’t know how many professors would buy her assertion that getting all 15 questions wrong demonstrates her knowledge, even though identical reasoning is used as proof of knowing correct answers when the probability of getting all the questions correct is about the same. (In this example, the probability of getting all 15 questions correct just by guessing is (.20)¹⁵, which is a number far less than .0001.) Personally, if I were the professor, I would give Sara high marks for creativity and for her understanding of statistical principles. It is possible that Sara did know “something” about the topic, but that “something” was systematically wrong. I would also point out to her that it is possible that she was both unprepared and unlucky enough to guess wrong fifteen times. After all, unusual events do happen sometimes.

Conjunction Error—Applying the “And Rule”

The following problem was posed by Tversky and Kahneman (1983, p. 297):

Linda is 31 years old, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

For the following list of statements, estimate the probability that it is descriptive of Linda.

A. Linda is a teacher in elementary school.

B. Linda works in a bookstore and takes yoga classes.

C. Linda is active in the feminist movement.

D. Linda is a psychiatric social worker.

E. Linda is a member of the League of Women voters.

F. Linda is a bank teller.

G. Linda is an insurance salesperson.

H. Linda is a bank teller and is active in the feminist movement.

Stop now and estimate the probability for each statement.

* * *

The short paragraph about Linda was written to be representative of an active feminist, which is statement C. Thus, if we rely on common stereotypes of the “typical feminist,” C would seem to be a likely description. Look at statements F (bank teller) and H (feminist and a bank teller). How did you rank these two sentences? Most people believe that H is more probably true than F. Can you see why F must be more likely than H when being a bank teller and being a feminist are independent of each other? There are some bank tellers who are not active in the feminist movement. When determining the probability of both of two events occurring, you multiply the probabilities of each one occurring (the “and rule”). Thus, the probability of two events both occurring must be less likely than the probability of one of these events occurring. In Tversky and Kahneman’s study, 85% of the subjects judged statement H to be more probable than statement F. The error of believing that the occurrence of two events is more likely than the occurrence of one of them is called the conjunction error.

For those of you who think better with spatial arrays, let’s represent this problem with circle diagrams, a form of representation that was used with syllogisms in the chapter on reasoning. Draw one circle to represent every bank teller in the world, and a second circle to represent every feminist. The two circles have to overlap somewhat because there are some bank tellers who are feminists. This area of overlap is shaded in Figure 7.3. As you can see in Figure 7.3, the shaded area that represents all people who are both bank tellers and feminists must be smaller than the circle that represents all bank tellers because there are bank tellers who are not feminists.

Figure 7.3 The two circles represent “All Feminists” and “All Bank Tellers.” The intersection of these two circles shows those individuals who are both feminist and a bank tellers. Because there are feminists who are not bank tellers and bank tellers who are not feminists, the area of their overlap must be smaller than either set alone.

Now that you understand the conjunctive error, try the next question (also taken from Tversky & Kahneman, 1983, p. 308):

A health survey was conducted in a sample of adult males in British Columbia, of all ages and occupations.

Please give your best estimate of the following values:

What percentage of the men surveyed have had one or more heart attacks? ____

What percentage of the men surveyed both are over 55 years old and have had one or more heart attacks? ____

Stop now and fill in the blanks above with your best estimate of these values.

Over 65% of the respondents believed that a higher percentage of the men would be both over 55 and have had a heart attack than the percentage of men who reported that they had a heart attack. Do you recognize this as another example of a conjunction error? The probability of two uncertain events both occurring cannot be greater than the probability of just one of them occurring.

There are some psychologists who object to all of the emphasis on the ways people make mistakes in their thinking (Gigerenzer, Todd, & the ABC Research Group, 2000; Gigerenzer, 2007). They argue that most of us think pretty well, given real-life circumstances, which include limited time to think through problems and limited information. Most of the time, we rely on heuristics, which are “rules of thumb” or shortcuts for thinking. Gigerenzer and his colleagues consider problems like the “Linda” problem to be “tricks.” In this case, we are asked to make a decision about one person, when the probability data refer to the population of all bank tellers and all feminists. He argues that it is not irrational to use “typicality information” (e.g., the typical feminist has certain traits) when making a “best guess” about one person.

There has been much heated debate over the “Linda” problem, probably because so many of us get it wrong. Gigerenzer (2007, p. 97) argues that the word “and” is sometimes used to signify a chronology of events and when used this way, it conveys a different message. Consider this example:

Mark got angry and Mary left him. The implication is that Mary left him after and probably because he got angry.

Or this example:

Verona is in Italy and Valencia is in Spain.

This is identical in meaning to:

Valencia is in Spain and Verona is in Italy.

The underlying idea with these examples is that the word “and” does not always signal the conjunction of two events and that is why so many people get the “Linda” problem wrong. Others argue that regardless of the fact that “and” does not always signal conjunctions, it does in the “Linda” problem, and most people get it wrong. This is an area where psychologists disagree about what is rational or reasoned thinking, and it is likely to continue to be a hotly debated topic in the next several years. The point made by Gigerenzer and others who are more optimistic about the quality of human cognition is certainly valid. Many of the examples that show how poorly most of us think seem artificial or lack important information that we normally would consider. Like many of psychology’s debates, each side seems to have part of the answer. Under some circumstances, people are easily misled and thinking goes wrong in predictable ways, but under other circumstances, people can think fairly well. The situation is like the proverbial glass that is both half-full and half-empty. Regardless of your personal opinion about this debate, it is true that with hard work, we can all learn to think better—a fact that should make both the optimists and pessimists very happy.

Cumulative Risks—Applying the “Or Rule”

It should be obvious that the probability of getting three questions correct by chance when there are five alternatives will be much smaller than the probability of getting just one question correct by chance, and the probability of getting at least one question correct by chance out of three questions will be higher than the probability of getting one question correct when there is only one question. The kinds of examples presented so far were deliberately simple. Let’s see how this principle applies in real world-settings.

Most real-life risks involve repeated exposure to a risky situation. Consider driving. The probability of having an accident in one car ride is very low. But what happens to this probability when you take hundreds or thousands of car rides? According to the “or rule,” this is the probability of an accident on the first or second or. … n^th car ride. In an interesting study of how people understand the concept of cumulative risk, Shaklee (1987) gave subjects different probability values that supposedly corresponded to the yearly risk of a flood. The subjects then had to estimate the likelihood of a flood in one month, five years, 10 years, and 15 years. Only 74% of her subjects knew that the likelihood of a flood increased with intervals over one year. Among those who gave higher probability values for intervals over one year, most seriously underestimated the cumulative probability.

The message here should be clear: When you are determining risk, it is important to understand whether the value you are being given is per some unit of time (e.g., one year) and how cumulative risks increase with repeated exposure. It seems that many people do not understand the concept that cumulative risks are greater than one-time risks.

Expected Values

Which of the following bets would you take if you could only choose one of them?

1. The Big 12

It will cost you $1 to play. If you roll a pair of dice and get a 12, you will get your $1 back, plus another $24. If you roll any other number, you will lose your $1.

2. Lucky 7

It will cost you $1 to play (same cost as above). If you roll a “lucky 7” with a pair of dice, you will get your $1 back, plus another $6. If you roll any other number, you will lose your $1.

Stop now and select either 1 or 2.

Most people choose 1, reasoning that $24 if a 12 is rolled is four times more than they can win if a 7 is rolled, and the cost is the same for each bet. Let’s see if this thinking is correct.

In order to decide which is the better bet, we need to consider the probability of winning and losing and the corresponding value of each. There is a formula that will take these variables into account and yield the expected value (EV) for each gamble. An expected value is the amount of money you would expect to win on each bet if you continued playing over and over. Like all probability values, it depends on what would happen in the long run. The formula for computing an expected value (EV) is:

EV = (probability of a win) × (value of a win) + (probability of a loss) × (value of a loss)

Let’s consider the EV for Choice 1. We will begin by computing the probability of rolling a 12 with a pair of dice. This is shown in Figure 7.4.

There is only one way to roll a 12, and that is with a 6 on each die. The probability of this happening when the dice are fair is 1/6 × 1/6 = 1/36 = .028. (Because we are interested in finding the probability of a 6 on the first and the second die, we use the “and rule” and multiply.) Thus, we would expect to roll a 12 about 2.8% of the time. What is the probability of not rolling a 12? Because we are certain that you will either roll a 12 or not roll a 12 (this covers all possible events), you can subtract .028 from 1.00. The probability of not rolling a 12 is .972. (You could also arrive at this figure, with some small rounding differences, by calculating the probability of each of the 35 other possible outcomes—each will be 1/36—and adding them together.)

Using these probability values, the EV formula for Choice 1 becomes:

(Choice 1):

EV = (Probability of a 12) × (Value of a 12) +

(Probability of not getting a 12) × (Value of not getting a 12)

EV = [(.028) × ($24)] + [(.972) × (-$1)]

EV = $.672 -$.972

EV = -$.30 for (Choice 1)Let’s review what happened in this formula. If you rolled a 12, you would win $24, which is the value associated with this win. If you rolled a number other than a 12, you would lose the $1 you paid to play this game, thus -$1 is the value associated with this loss. The probability of a win was multiplied by the value of a win. The probability of a loss was multiplied by the value of a loss. Then, these two products were added together. The EV of this bet, -$.30, means that in the long run, if you continue playing this game many times, you could expect to lose, on the average, $.30 for every game played. Of course, on any single game, you would either lose $1 or win $24, but after many, many games you would have lost an average of $.30 per game. If you played 1,000 games, making the same bet each time, you would be $300 poorer.

Figure 7.4 A tree diagram depicting all possible outcomes of rolling a pair of dice.

How does this compare with Choice 2? To calculate the EV of Choice 2, we will begin by computing the probability of rolling a 7. How many ways is it possible to roll a 7 with a pair of fair dice? You could get a 1 on one die and a 6 on the other; a 2 and a 5; a 3 and a 4; a 4 and a 3; a 5 and a 2; or a 6 and a 1. Thus, there are 6 possible ways to roll a 7 out of 36 possible outcomes. The probability of any one of these outcomes is 1/6 × 1/6, which equals 1/36. (This is the probability of rolling, for example, a one on the first die and a six on the second die.) Thus, to determine the probability of one number followed by a second number, you would apply the “and rule.” Because you are now concerned with the probability of a 1 followed by a 6 or a 2 followed by a 5 or a 3 followed by a 4 or a 4 followed by a 3 or a 5 followed by a 2 or a 6 followed by a 1, you should recognize the second step as a case where the “or rule” is needed. Since there are six possible combinations, you would add 1/36 six times (which is, of course, the same as multiplying it by 6). Thus, the probability of rolling a 7 with a pair of dice is 6/36 (1/6 or .167). The probability of not rolling a 7 is 1—.167, which equals .833. Now we calculate the EV of Choice 2.

(Choice 2):

EV = (Probability of a 7) × (Value of a 7) +

(Probability of not getting a 7 × (Value of not getting a 7)

EV	= [(.167) × ($6)] + [(.833) × (-$1)]
	= ($1.002—.833)
	= $.169, or approximately $.17 for (Choice 2)

This means that if you continued to gamble on Choice 2, you would win an average of $.17 for every game played. Thus, after 1,000 games, you could expect to be $170 richer. Of course, as in Choice 1, you would never actually win this amount on any single game; this is what would result if you continued playing many, many games. This is what would happen “in the long run.”

Even though you might have originally thought otherwise, Choice 2 is the better choice because of the relatively high probability associated with rolling a 7. Seven has a high probability because there were six possible combinations that would add up to a 7.

There is a party game that is based on the principle that the more ways an event can occur, the more likely it is to occur. Suppose you get a random sample of 40 people together in a room. Estimate the probability that two of them share the same birthday. You may be surprised to learn that the probability is approximately .90. Can you figure out why it is so high? There are many, many ways that 40 people can share the same birthday. To figure out the exact probability, you would take all combinations of 40 people, two at a time. Thus, we would have to start with the combination of person 1 with person 2, then person 1 with person 3, and so on until person 1 is matched with person 40; then we would begin again matching person 2 with person 3, 2 with 4, on until 2 is matched with 40. This whole process would have to be repeated until every one of the 40 people is matched with every other one. Because there are so many possible combinations of any two people sharing any birthday in the year, this “coincidence” is more probable than it may have seemed at first. The probability of two people sharing a common birthday is over .50 when there are 23 people and over .75 when there are 32 people (Loftus & Loftus, 1982). You can use this knowledge to make wagers at parties or at any gathering of people. It is a fairly good bet when the number of people is close to 40. Most people find it hard to believe that the probability is so high.

You can also use your knowledge of probability to improve your chances in other situations. Take, for example, Aaron and Jill, who have been arguing over who should take out the garbage. Their mother agrees to help them settle this matter by picking a number from 1 to 10. The one whose number comes closer to the one selected by their mother will win the dispute. Aaron goes first and picks “3.” What number should Jill select to maximize her chances of winning? Stop now and decide what number she should select. The best number for Jill to pick is 4. If her mother were thinking of any number greater than 3, Jill would win with this strategy. Thus, she can change the probability of winning in what seems like a chance situation.

Subjective Probability

We usually do not deal directly with known or objective probabilities, such as the probability of rain on a given day or the probability of developing heart disease if oral contraceptives are taken. Yet, every day we make decisions using our best estimate of the likelihood of events. Subjective probability refers to personal estimates of the likelihood of events. This term is in distinction from objective probability, which is a mathematically determined statement of likelihood about known frequencies. Subjective probabilities are often wrong in two ways. First is the best guess or subjective estimate of the probability of an event. Consider Charlie, whose dating hopes are described below. More than anything in the world, he wants a kiss from the fair Louise. As you read below, he estimates the probability that she will agree to go on a date with him as .10 and the subsequent probability that she will kiss him as .95. He estimated these probabilities, and he could be way off.

A second way that subjective probabilities can be in error is in his estimate of how high is the “value” of this kiss. There is a large body of research literature on affective forecasting, which refers to our ability to predict (forecast) how we will feel (affect) about an event (Hoerger, Quirk, Lucas, & Carr, 2010). As you can probably guess, most people are not very good at accurately predicting how they will feel about something in the future. Alas, Charlie may have overestimated the power of his first kiss. Psychologists who have studied subjective probability have found that human judgments of probability are often fallible, yet we rely on them to guide our decisions in countless situations.

Base-Rate Neglect

Charlie is anxious to experience his first kiss. If he asks Louise to go to the movies with him, he is only 10% sure that she will accept his invitation, but if she does, he is 95% sure that she will kiss him goodnight. What are Charlie’s chances for romance?

Initial or a priori probabilities are called the base rate. In this problem, the first hurdle that Charlie has to “get over” is getting Louise to go out with him. The probability of this occurring is 10%. It is important to think about this figure—the base rate. Ten percent is a fairly low value, so it is likely that she will not go out with him. He wants to know the probability of two uncertain events occurring—she both goes out with him and she kisses him. Before we start to solve this problem, think about the kind of answer you would expect. Will it be greater than 95%, between 95% and 10%, or less than 10%?

To solve this problem, we will use a tree diagram to depict the possible outcomes and the probability associated with each. Of course, it is not likely that Charlie or any other Romeo-wanna-be will actually compute the probability values or draw a tree diagram for determining the probability of this momentous event, but this example demonstrates how likelihoods combine. Maybe he will decide that the probability of a kiss from Louise is so small that he will opt for Brunhilda, who is both more likely to accept his offer of a date and succumb to his romantic charms. Besides, anyone who has actually estimated probability values for romance might also want to be more precise in his estimates of two or more events.

We will start with a tree diagram that first branches into “Louise accepts his date” and “Louise declines.” A second branch will be drawn from the Louise accepts his date node indicating whether he gets kissed or not. Each branch should have the appropriate probability labels. Of course, if Louise declines his invitation, Charlie definitely will not get kissed. The branch from the “Louise declines” node is thus labeled 1.00 for “Charlie doesn’t get kissed.”

According to the and rule for finding the probability of two (or more) events, the probability that Louise will kiss Charlie goodnight is:

.10 × .95 = .095

Are you surprised to find that the objective probability is less than the low base rate of 10% and much less than the higher secondary or subsequent rate of 95%? Most people are. Hopefully, you recognized that any value greater than 10% would have been indicative of a conjunction error. Recall from the earlier section on conjunction errors that the probability of two uncertain events both occurring (Louise accepts and kisses Charlie) must be less than the probability of either event alone. Most people ignore (or underestimate) the low base rate and estimate their answer as closer to the higher secondary rate. In general, people tend to overestimate the probability of two or more uncertain events occurring. This type of error is known as base-rate neglect. Of course, in real life, estimates of probabilities for events like having a date accept your invitation and then the probability that she or he will kiss you are not very exact, so it may be a bit of a stretch to ask poor Charlie, in this example, to compute the combined probabilities of both of these events actually happening. Other situations in which probabilities are calculated from population values, like the “probability of getting malaria if you travel in a tropic country during a rainy summer” and the “probability of having a reaction to malaria medicine” may be a more realistic example where you might want to calculate joint probabilities. But, in either case, it is important to understand that if one event occurs rarely, the likelihood of that event plus another one will be even less than the rare one alone.

Gambler’s Fallacy

The “Wheel of Fortune” is a popular game at fairs, casinos, amusement parks, and on television game shows. It consists of a large wheel, which can be spun. The wheel is divided into many numbered sections, much like a roulette wheel. A rubber marker indicates the winning number.

Suppose that your friend, Vanna, decides to approach the Wheel of Fortune in a scientific manner. She sits at the Wheel of Fortune and records when each number comes up as the winning number. Suppose Vanna has recorded the following series of winning numbers: 3, 6, 10, 19, 18, 4, 1, 7, 7, 5, 20, 17, 2, 14, 19, 13, 8, 11, 13, 16, 12, 15, 19, 3, 8. After examining these numbers very carefully, she proclaims that a 9 has not appeared in the last 25 spins; therefore, she plans to bet heavily on number 9, because it is now much more likely to occur. Would you agree with her that this is a good bet?

If you responded, “Yes,” you have committed a very common error in understanding probability. The Wheel of Fortune has no memory for which numbers have previously appeared. If the wheel had been built so that each number is equally likely to win, then a 9 is equally likely on each spin, regardless of how frequently or infrequently it appeared in the past. People believe that chance processes like spinning a Wheel of Fortune should be self-correcting so that if an event has not occurred in a while it is now more likely to occur. This misconception is called Gambler’s Fallacy. (Of course, if you recall the roulette example, there are some likely imperfections in the wheel which would make some numbers more likely than others, but if this were so, then the fact that 9 was not a recent outcome would mean that it is less likely to occur in a future spin—a prediction that is opposite from gambler’s fallacy.)

Gambler’s fallacy can be found in many settings. Consider a sports example. A “slumping” batter who has not had a hit in a long while is sometimes believed to be more likely to have a hit because he is “due” one. A sports enthusiast, who is a friend of mine, told me the following story about Don Sutton, a former pitcher for the Dodgers. One season, Sutton gave up a great many runs. He predicted that this “slump” would be followed by a “correction” so that he would end up the season at his usual average. Unfortunately, there is no correction for chance factors and, because he had such a poor start to the season, he ended the season below his usual average.

Often, people will continue to believe in Gambler’s Fallacy even after it has been explained to them. Students have told me that while they can understand on an intellectual level why Gambler’s Fallacy must be wrong, on an intuitive or “gut” level, it seems that it ought to be right. Understanding probability often requires that we go against our intuitive hunches because they are often wrong. Let’s try another example.

Wayne and Marsha have four sons. Although they really do not want to have five children, both have always wanted a daughter. Should they plan to have another child, because they are now more likely to have a daughter, given that their first four children were boys? If you understand Gambler’s Fallacy, you will recognize that a daughter is as likely as a son on the fifth try, just as it was on each of the first four. (Actually, because slightly more boys than girls are born, the probability of having a boy baby is slightly higher than the probability of having a girl baby.) Gambler’s Fallacy also has a flip side, the belief that random events run in streaks. Consider the following two scenarios:

A.	A basketball player has just missed 2 or 3 of the last shots in a row. She is about to shoot again.
B.	A basketball player has just made 2 or 3 of the last shots in a row. She is about to shoot again.

Is she more likely to make the basket in A or B?

Gilovich (1991) asked questions like this one to knowledgeable basketball fans and found that 91% believed that the player was more likely to be successful in B than in A. In other words, they believed that players shoot in streaks. In order to determine whether there is any evidence to support the “belief in streaks,” Gilovich analyzed data from the Philadelphia 76ers. He found that:

• if a player just made a shot, 51% of the next shots were successful;

• if a player just missed a shot, 54% of the next shots were successful;

• if a player made 2 shots in a row, 50% of the next shots were successful;

• if a player missed 2 shots in a row, 53% of the next shots were successful.

These data show no evidence that players shoot in streaks. Yet, interviews with the 76ers players themselves showed that they believe that they shot in streaks. It is very difficult to convince people that chance is exactly that—it doesn’t self-correct or run in nonrandom streaks.

Making Probabilistic Decisions

Most of the important decisions that we make in life involve probabilities. Although decision making is discussed more fully in the next chapter, let’s consider how tree diagrams can be an aid for decision making.

Edith is trying to decide on a college major. She attends a very selective university that has independent admissions for each of its majors. She is seriously thinking about becoming an accountant. She knows that the Accounting Department accepts 25% of the students who apply for that major. Of those accepted, 70% graduate, and 90% of those who graduate pass the national accounting examination and become accountants. She would like to know what her chances are of becoming an accountant if she pursues the accounting major.

To answer this question, draw a tree diagram with branches that represent the “path” for success in accounting.

From the above diagram, you can see that the probability of becoming a successful accountant is equal to .25 × .70 × .90, which is .158. At this point, Edith might want to consider other options. For example, she could consider applying to both the accounting major and to the education major. She could recalculate her chances for success in one of these majors, both of these majors (if this is a possible option for her), and neither of these majors.

This example assumes that we have no additional information on which to base Edith’s chances of success. Suppose instead that we know that Edith has excellent math skills. Shouldn’t this sort of information change the probabilities involved and make it more likely that Edith be admitted, graduate, and succeed in a math-related occupation? Intuitively, the answer is “yes.” Let’s see how the problem changes by considering José’s probability for success in the following example.

Combining Information to Make Predictions

José has always wanted to be an actor. Accordingly, he plans to sell his worldly possessions and head for a career in the “Big Apple” (the loving nickname for New York). Suppose that you and José both know that only about 4% of all aspiring actors ever “make it” professionally in New York. This value is the base rate; it is based on information that is known before we have any specific information about José. Stop and think about this figure—the base rate. It tells us that very few aspiring actors become professionals in this field. In other words, the chance of success is low. Suppose that you had no additional information about José. What value would you predict as his chance of success? If you said 4%, right on! In the absence of any other information, use the base rate.

José tells you not to worry, because 75% of those who are successful have curly hair and can sing and tell jokes well. Because he has curly hair and is a good singer and a hilarious comedian, he feels confident that he will soon be sending 8 × 10 glossy pictures of himself to his fan club members. This second value is called the secondary; it is the probability value that relates specific information about characteristics that are associated with José and with an outcome. We will use these two probability values to decide if José’s optimism is warranted. Exactly how likely is he to succeed? Before you continue, make a probability estimate for his chance of success. Remember, probabilities range from 0 to 1, with 0 meaning he will definitely fail and have to return to Peoria, and 1 meaning he will definitely succeed on Broadway. Stop now and make a subjective probability judgment of his chance for success.

Can you think of a way of objectively finding his chance of success? In order to arrive at an objective probability, you will need to know another number, one that is often ignored—the percentage of those who fail and have the attributes that are associated with success (in this case, curly hair and the ability to sing, dance, and joke). Few people realize that they need to consider this value in assessing the probability of success. For ease of reference, I will call the attributes that are associated with success (curly hair and ability to sing and tell jokes) “curly hair,” and the absence of these attributes, I will call “not curly hair.” Suppose that 50% of those who fail have these attributes. Once again, tree diagrams can be used to determine probabilities in this context. Let’s begin at a starting point and consider all possible outcomes. In this case, he will either succeed or fail, so we will label the first branches “succeed” and “fail.” As before, we will put the probability of each event along the appropriate branch:

These two probabilities (.04 and .96) sum to 1.0 because they include all possibilities. One of these two possibilities must occur, so they will add up to 1.0 to indicate absolute certainty.

José knows that 75% of those who succeed have curly hair. In this example, what we are trying to find is the probability of a certain outcome (success) given that we already have information that is relevant to the probability of that outcome. Let’s add a second branch to the tree diagram, branching off from the succeed node and the fail node. There are four different probabilities involved in this example: the probability of succeeding and having curly hair, the probability of succeeding and not having curly hair, the probability of failing and having curly hair, and the probability of failing and not having curly hair. These four possibilities are shown in the next tree diagram.

Because 75% (.75) of those who succeed have curly hair and 25% (.25) do not have this attribute, the sum of all probabilities from a tree node must sum to 1.0. Similarly, 50% of those who fail have curly hair, while 50% of those who fail do not possess this attribute. Because we are considering everyone who fails, these values must also sum to 1.0.

Once the tree diagram is drawn, it is a simple matter to compute José’s objective probability of success. As before, multiply along each branch to find the probabilities. In this case, we would multiply the values along each branch of the tree diagram and compile the information in a chart:

P[Succeed & Curly Hair]	= .04 × .75 =	.03
P[Succeed & Not Curly Hair]	= .04 × .25 =	.01
P[Fail & Curly Hair]	= .96 × .50 =	.48
P[Fail & Not Curly Hair]	= .96 × .50 =	.48
		1.0000

To determine José’s true chance of success, we need to divide the proportion of those who succeed and have curly hair (.03) by the total proportion who have curly hair (.48 + .03 = .51). We are trying to predict José’s success based on the knowledge that he has curly hair and that some proportion of all people with curly hair are successful. Of all of those with curly hair (.51), what proportion of them succeed (.03)?

Thus, José’s chances for success are 50% higher (6% versus 3%) than they are for any unknown, aspiring actor, but they are still very low. Knowing that he has certain attributes that are associated with success improved his probability of success above the base rate, but the improvement was very small.

You may find it easier to follow the logic of these calculations by putting the information in a table format.

	Succeed	Fail	Row Totals
Curly hair, etc.	0.03	0.48	0.51
No curly hair, etc.	0.01	0.48	0.49
Column Total	0.04	0.96	1.00

Are you surprised to find that his chance of success is so low given that the posterior or secondary probability value was so high (75%)? Most people are. The reason that José has such a slim chance of becoming an actor is because so few people, in general, succeed. The probability value José obtained was close to the a priori, or base rate, of success among all aspiring actors. Because so few actors, in general, succeed, José, and any other would-be thespian, has a low chance for success. In general, most people overestimate success when base rates are low and underestimate success when base rates are high. In the earlier example concerning Edith, we had only base-rate information to use in predicting success. By contrast, we had additional information about José that allowed us to improve upon the base rate when predicting his success, although because of the low rate of success for actors in general, the improvement was slight.

For those of you who prefer to think spatially or think in terms of frequencies instead of probabilities, think about a group of 100 people, 4 of whom are successful actors (4%) and 96 are not (96%). This group is shown in Figure 7.5. Four of the 100 “people” depicted are smiling—these represent the successful actors. If you had no other information to use to predict success for José, you would use this base rate information and give him a 4% probability for success.

Now let’s consider the additional information. Three of the 4 of those who are shown smiling have curly hair (3/4 or 75%), whereas half (50%) of those who are not smiling have curly hair. This information is combined with the base-rate information. It is depicted in Figure 7.6 with the addition of curly hair to the successful and unsuccessful actors. Of the 4 smiling faces, 3 have curly hair (75%), and of the 96 frowning faces, 48 (50%) have curly hair.

By examining these figures, it should be easy to see that what we are doing mathematically is finding the proportion of smiling faces with curly hair relative to all of the faces with curly hair in order to use this information about José to predict his success. Graphically, this is 3 smiling faces with curly hair as a proportion (or fraction) of the 51 faces that have curly hair:

3/51 = 0.06

To review, when you are calculating the probability of an outcome, given that you have information that is relevant to its probability, you will:

Figure 7.5 Pictorial representation of a 4% success rate. Note that 4% of the faces are smiling, which represents the base rate—4% of all aspiring actors succeed. If we had no personal information about José, his expected probability of success would be 4%.

1. Draw a complete tree diagram, with the base rate information (e.g., success or failure) as the first set of nodes. Use the secondary information to draw the second set of nodes.

2. Make a chart with all combinations of the base rate information and secondary information as the rows in the chart.

3. Multiply probabilities across each of the branches of the tree diagram and fill in each row of the chart with these values.

4. Form a ratio (fraction) with the probability value from the branch that you are interested in (e.g., success given that he has curly hair) as the numerator, and the sum of this value and the other branch that contains the same conditional statement (e.g., failure given that he has curly hair). In this example, 51 people had curly hair (3 were successful and 48 were not successful), so that is the denominator.

5. Check your answer. Does it make sense? Would you expect, as in this example, that the probability of success is higher than the base

Figure 7.6 Pictorial representation of the relative proportion of successful and unsuccessful actors who have the same attributes of José. These are depicted by the addition of curly hair. When we use personalized information to make predictions, we form a fraction of the proportion of those who are successful and have the attribute (smiling and curly hair) over the total proportion of those who have the attribute (curly hair). In this case, 3 smiling faces with curly hair form the numerator and the 51 faces that have curly hair form the denominator.

rate because we know something about the individual that is associated with success? (If we knew that José had some trait that was associated with failure, we would predict that his chance of success would be less than the base rate, but with low base rates, it probably will not be much less.)

Let’s try a more realistic example: There are many diseases that occur with low base rates in the population. Medical test results must be interpreted in light of the relevant base rates for each disease. Medicine, like most other disciplines, is a probabilistic science, yet few physicians receive training in understanding probabilities. Failure to utilize base-rate information can lead to improper diagnoses. Base rate neglect is a pervasive error in thinking about probabilistic outcomes. Consider Dreman’s (1979) summary of a large body of research on this effect:

The tendency to underestimate or altogether ignore past probabilities in making a decision is undoubtedly the most significant problem of intuitive predictions in fields as diverse as financial analysis, accounting, geography, engineering, and military intelligence. (cited in Dreman, D., 1998, p. 236)

Thinking with Frequencies

A common response to problems like the one with José is that the problem is too difficult. In fact, even though many physicians have to interpret laboratory tests where the information is similar to that in the José problem, many of the physicians admit that they do not understand how they can use probability information to make important decisions about the health of their patients. Many people find it easier to think about probabilities in situations like this one by imaging frequencies (Gigerenzer & Hoffrage, 1995; Lovett & Schunn, 1999). Here is a real-life example that was used by Gigerenzer and his colleagues who looked for ways to make computing probability information easier to understand. Everyone was given the same opening paragraph:

To facilitate early detection of breast cancer, women are encouraged from a particular age on to participate at regular intervals in routine screening, even if they have no obvious symptoms. Imagine you conduct in a certain region such a breast cancer screening using mammography. For symptom-free women aged 40 to 50 who participate in screening using mammography, the following information is available for this region:

After reading this paragraph, half of the physicians in the study were given the probability format and half were given the frequency format. Here is the same information in two different formats (Gigerenzer & Hoffrage, 1995):

Probability format: The probability that one of these women has breast cancer is 1%. If a woman has breast cancer, the probability is 80% that she will have a positive mammography test. If a woman does not have breast cancer, the probability is 10% that she will still have a positive mammography test. Imagine a woman (aged 40 to 50, no symptoms) who has a positive mammography test. What is the probability that she actually has breast cancer?
____________? (fill in your answer)

Frequency Format: Ten out of every 1000 women have breast cancer. Of these 10 women with breast cancer, 8 will have a positive mammography test. Of the remaining 990 women without breast cancer, 99 will still have a positive mammography test. Imagine a sample of women (aged 40 to 50, no symptoms) who have positive mammography tests. How many of these women actually have breast cancer? ____ out of ____?

How did you go about answering these questions?

For the probability format:

The researchers found that when this information was given in the probability format, only 10% of the physicians could reason correctly, but when the information was given in the frequency format, 46% reasoned correctly. The ability to combine probability information is an important thinking skill—decisions about treatment options depend on how people understand numbers. With practice, we can all think better with numbers.

The Problem of False Positives

People may not like or even understand what scientists say, especially when what they say is complex, counterintuitive or ambiguous.

—John Allen Paulos (2009, December 10, para. 1)

The public outcry against two recent recommendations from national health organizations shows the difficulty in understanding information about health risks. Scientists at the U.S. Preventive Services Task Force (USPSTF, 2009) have recommended against routine mammograms for women in their 40s who have no symptoms of cancer. A similar recommendation for men was issued by the same group in 2011; they recommended against a commonly used test (PSA, which stands for protein-specific antigen) that screened for prostate cancer. These recommendations are counterintuitive because most people reason that early screening increases the likelihood that cancer will be detected while in an early stage and early detection can save lives. How can responsible scientists make these recommendations?

The scientists determined that these early screening tests do more harm than good, but it will require some work to explain the reasoning behind this decision. I rely on the excellent article on the prostate screening controversy by Arkes and Gaissmaier (2012) for this explanation. First is a concept that comes up repeatedly in this book—anecdotes are powerful. The statistical analysis behind the USPSTK report does not have the same impact as learning that the brother-in-law of your karate teacher is alive today because he has a PSA test that identified prostate cancer. The limitations of anecdotes are repeated throughout this book.

To explain the reasoning behind NOT getting screened for cancer, Arkes and Gaissmaier (2012) ask you to imagine two large rooms each containing 1,000 men. The men in the Screened Room all had a PSA test, and those in the Not Screened Room did not. About eight men from each room will die from prostate cancer in the next 10 years (Djulbegovic et al., 2010). But how is that possible if many men claim that the PSA test saved their lives?

Now imagine three groups of men in the Screened Room.

Group 1: The PSA Test detected prostate cancer, which led to early treatment, and these men are alive today. The data are not clear as to whether or not their lives were really extended by early detection and treatment, but the men in this group believe their lives were saved by the PSA Test.

Group 2: The second group consists of approximately 20 men who had positive identifications of cancer from the PSA Test, and they received early treatment. But, for the men in this group, the cancers would not have done any harm (they had a slow growing type of cancer) if it had not been detected. Unfortunately many of these men have serious side effects from the treatment, including impotence and/or incontinence. They also had a small risk of dying from the treatment. They do not know that they would have been better off without the PSA Test because the treatments were unnecessary and harmful, so they erroneously believe that the test saved their lives.

Group 3: There are about 180 men in the Screened Room with false positive tests. This means that the test was wrong—it diagnosed prostate cancer when they did not have cancer. They underwent biopsies (with some small risk in addition to the pain and expense). Most of these men are relieved to find out that they really do not have cancer and few will be angry about the false positive test result.

With this thought experiment, we can understand why many men who were screened for prostate cancer believe that it is a good test. They do not know about the men in the Unscreened Room where the death rate from prostate cancer is identical to that for men who were screened. Of course, in real life, these rooms do not exist, and we do not have the information about screening presented in a way that is easy to understand. The real situation is actually even more negative when considering cancer screening. Let’s assume that 6% of the population of all men actually has prostate cancer. This is a low base rate—which means that it occurs with a low probability. When the base rate is low (as seen in the example presented earlier with the low probability of becoming a successful actor), there will be many false positives (men who are diagnosed with cancer and do not have it) and false negatives (men with cancer who are not identified with the test). Unfortunately, many people have an emotional need for certainty, especially when dealing with health information (Gigerenzer, Gaissmaier, Kurz-Milcke, Schwartz, & Woloshin, 2007). Traditionally, doctor–patient relationships were based on trust, not statistics, but by understanding basic principles in likelihood and uncertainty, we can all make better choices and live better lives.

Nonregressive Judgments

Harris is a new student at Rah-Rah State University. The average grade point average (GPA) for all students at Rah-Rah is 2.8. Harris is new to this college and has not yet taken any exams. Although we have no information about Harris specifically, what would be your best guess that his grade point average will be?

Stop here and make your best guess for his grade point average.

After his first set of midterm exams, Harris has a midterm GPA of 3.8. Given this new information, what would you now predict for Harris’ GPA at the end of the school year? Most people readily answer the first question with 2.8, the average GPA for all students at Rah-Rah. This is a correct answer, as in the absence of any specific information, the average for all students, the population at Rah-Rah, is the best estimate for anyone. Most people answer the second question with 3.8. Unfortunately, this is not the best answer. Although it is true that someone who scores high on midterms will also tend to score high on finals, the relationship is not a one-to-one or perfect relationship. In general, when someone scores extremely high on some scale, she or he will score closer to the average the second time. Thus, the best prediction for Harris’ GPA at the end of the school year will be less than 3.8 and greater than 2.8. (The actual value can be mathematically determined, but the calculations are beyond the scope of this book.) This is a difficult concept to understand because most people find it to be counterintuitive.

It may be useful to think about a sports example. Consider your favorite athletes. Although they may have a truly exceptional performance one day, most often they will perform closer to average, but still above average on other days. After all, no one bowls all perfect games or bats 1,000. Sports enthusiasts will recognize this principle as the “sophomore slump.” After an outstanding first year at a sport, the star will usually perform closer to average during her second year. This phenomenon is called regression toward the mean. (Mean is just another term for average—it is computed by adding up all the values you are interested in and dividing that sum by the number of values.)

Earlier in this chapter, I talked about the Laws of Chance. No one can predict accurately the height of any particular individual. But, in the long run, that is with many, many extremely tall fathers, most of their son’s heights will show regression toward the mean. Thus, as before, we can make better predictions by knowing about the Laws of Chance, but we will not always be accurate. It is important to understand this concept whenever dealing with probabilistic events.

Kahneman and Tversky (1973) studied what can happen when regression toward the mean is not understood by professionals. Israeli flight instructors were told that they should praise their students when they successfully performed difficult flight patterns and maneuvers and that they should criticize exceptionally poor performance. Based on what you have just learned about regression toward the mean, what, in general, should happen after a pilot performs extremely well? Subsequent maneuvers should be closer to average, or less than exceptional, because the performance moved or regressed toward the mean (average). Conversely, what should you expect to happen following very poor performance? Again, subsequent maneuvers should tend to be more average, or in this case, they would improve, although they may still be less than average. The Israeli flight instructors did not understand regression toward the mean and erroneously concluded that praise led to poorer performance and criticism improved it.

Regression to the mean is ubiquitous, yet few people ever recognize it. Let’s consider another example of regression to the mean. Suppose that you learn of a self-help group for people with children who seriously misbehave. (There really are such groups.) Most parents will enter these groups when their child’s behavior is at its worst because most parents will seek help only when the behavior is extremely bad. After a few weeks in the program, many parents report an improvement in their child’s behavior. Can we conclude that the program probably worked to help parents control their children’s behavior? Think regression to the mean! If parents entered the program when the behavior is extremely bad, then no matter what they do, including nothing at all, the child’s behavior will most likely regress toward the mean. In other words, if a child is extremely misbehaved, it is statistically true that the child will move toward the average on measures of behavior. We would not predict angelic or even average behavior, just some improvement or movement toward the mean. Because this is a statistical prediction, sometimes, we will be wrong, but on the average and in the long run, we will be right with this prediction. Thus, we cannot conclude anything about the effectiveness of this program unless we conduct a true experiment of the sort that was described in the last chapter. We might randomly assign children and families to these groups and to “no treatment control groups” and then determine if the children in the self-help group were significantly better behaved than those who received no treatment. We must be able to randomly assign families to groups before we can conclude that the self-help group was helpful in improving the child’s behavior. Once you start looking for regression to the mean, you will be surprised how many events in life are best explained by “moving toward the average” and not to other causes.

Risk

We pride ourselves on being the only species that understands the concept of risk, yet we have a confounding habit of worrying about mere possibilities while ignoring probabilities, building barricades against perceived dangers while leaving ourselves exposed to real ones.

—Jeffrey Kluger (2006, para. 4)

If we examine data from thousands of communities around the United States or around the world, we will find that some communities have exceptionally high rates of some sorts of cancers, birth defects, brain tumors, unexplained deaths, and other maladies. How can we know if there are links between these high rates of illness and toxic substances, such as pesticides in the water, magnetic fields from electrical power lines, or chance?

The notion of frequency, or how often an event occurs, is inherent in the definition of probability. If an event is frequent, then its occurrence is highly probable. To determine the risk involved with a disastrous event, we need first to determine the frequency with which it occurs. Because most disastrous events are rare (e.g., plane crashes, leaks from nuclear plants) and, in some cases, take years before they are evident (e.g., cancers from environmental hazards), determining their frequency is a very difficult task. To understand how people make judgments involving risks, we need to understand how they determine the frequency of real-life risky events. Several researchers (e.g., Lichtenstein, Slovic, Fischoff, Layman, & Combs, 1978) have focused on the way people judge the frequency of lethal events. They studied these phenomena by asking college students and members of the League of Women Voters to decide which of two possible causes of death is more probable for several pairs of lethal events. To understand their experiment and their results, let’s try a few examples. For the following pairs of items, indicate which is the more likely cause of death and then estimate how much more likely your choice is than the other event:

A. asthma	or tornado
B. excess cold	or syphilis
C. diabetes	or suicide
D. heart disease	or lung cancer
E. flood	or homicide
F. syphilis	or diabetes
G. asthma	or botulism
H. poisoning by vitamins	or lightning
I. tuberculosis	or homicide
J. all accidents	or stomach cancer

[Researchers found that while, in general, people were more accurate as the differences in the true frequencies of occurrence between the events increased, they made a large number of errors in estimating the relative frequencies of the events. Their subjects overestimated the frequencies of events that occurred very rarely and underestimated the frequencies of those events that occurred very often. In addition, lethal events that had received a great deal of publicity (e.g., airplane crashes, flood, homicide, tornado, botulism) were overestimated, while those that were undramatic, silent killers (e.g., diabetes, stroke, asthma, tuberculosis) were underestimated. It seems that publicized events were more easily brought to mind, and this biased their judgments of frequency. Hazards that are unusually memorable, such as a recent disaster or an event depicted in a sensationalized way on the news, like a major plane crash or botulism in undercooked hamburgers, distort our perceptions of risk. This was clear when a few incidents of shootings in high schools in the United States were immediately followed by anxious parents who took their children out of public high schools and enrolled them in private schools. Of course, the shooting death of high school students was extremely tragic, but these same parents would allow their children to drive when they were sleepy or never instructed them about ways to prevent sexually transmitted diseases, both of which are far more likely to occur than a school shooting. The more salient events were perceived to be the greater risk than the less publicized but more likely dangers of an automobile accident or sexually transmitted disease. In Chapter 2, I made the point that memory is an integral component of all thinking processes. What we remember is a major influence on how we think.

Assessing Risks

How do the experts make decisions that involve potentially disastrous outcomes? How can we, as informed citizens and voters, make less risky decisions? Questions like these are timely, but not easy to answer.

The goal of risk assessment is to find ways to avoid, reduce, or manage risks. Risk is associated with every aspect of life. For example, approximately 200 people are electrocuted each year in accidents involving home wiring or appliances, and 7,000 people die each year in U.S. homes as a result of falls (most of these people are over 65). Yet, few of us would interpret these risks as great enough either to forgo electricity or stop walking in our homes. Other risks are clearly too large to take. Few of us, for example, would decide to cross a busy freeway wearing a blindfold. And still other risks are largely unknown, such as the release of a new chemical into the environment or the development of a new technology. Wilson and Crouch (1987) suggested several ways of estimating risks that voters and consumers should consider when deciding if an action or a technology is safe enough.

• One method of risk assessment involves examining historical data. For example, to understand the risk of cancer due to exposure to medical x-rays, there are data that indicate that for a given dose per year (40 mrem), there is an expected number of cancers (1,100). This sort of risk information can be compared to other known risks so that consumers can decide if the benefits of medical x-rays outweigh the risks.

• The risk of a new technology for which there are no historical data can be computed when the occurrence of the events are independent by calculating the risk of separate components and multiplying along the branches of a decision tree. This method of calculating probabilities was presented in an earlier section of this chapter. A well-known example is the probability of a severe accident at a chemical plant.

• Risks can also be calculated by analogy. (The use of analogies as an aid to problem solving is discussed more fully in the following two chapters.) When animals are used to test drugs, the experimenter is really using an analogy to extrapolate the risk to humans.

Biases in Risk Assessment

Understanding the role of optimistic biases in consequential and emotional domains such as health, relationships, and investments requires studying judgments in circumstances in which passions are strong. … We found that people are optimistic in their predictions—they judge preferred outcomes to be more likely than nonpreferred outcomes.

—Cade Massey, Joseph P. Simmons, and David A. Armor (2011, p. 279).

Psychologists and others who study the way people determine if something is “too risky” know that most of us fall prey to common biases when we assess the “murky psychometrical stew” (Paulos, 1994, p. 34) that constitutes the numerical information and misinformation that we need to interpret. Here are some common biases (Wandersman & Hallman, 1993):

1. When a risk is voluntary, it is perceived to be less risky than one that is not voluntary. For example, cosmetic surgery is often believed to be safer than a nonelective surgery. After all, patients choose to undergo cosmetic surgery, so they must rationalize that it is “safe enough.”

2. Natural risks are believed to be less hazardous than artificial ones. For example, many people believe that naturally occurring toxins in our food are less dangerous than ones that are caused by pesticides or preservatives that are added.

3. Memorial events in which many people are harmed at once are perceived as riskier than more mundane and less vivid events. An example of this effect is the large number of people who are terrified of plane crashes, but give little thought to automobile safety.

It is clear that personal risk perceptions are not the same as scientific risk estimates. Experts in risk assessment perceive risks based on annual mortality so that the event that results in the greater number of deaths is judged to be the greater risk. Experts, for example, ranked motor vehicles as riskier than nuclear power (because more people are expected to die from motor vehicle accidents), whereas samples of college students and members of the League of Women Voters ranked nuclear power as the greater risk (because it is an example of a spectacular dreaded outcome).

Answers to the questions about the probability of lethal events with the true frequency of each event (rate per 100,000,000) are found on the next page. Check your answer and see if you made the common errors of overestimating events that are more memorable and likely to affect many people at one time (like a plane crash) and underestimating those risks over which we believe have some control (like driving an automobile).

More Likely	Rate	Less Likely	Rate
A. Asthma	920	Tornado	44
B. Syphilis	200	Excess Cold	163
C. Diabetes	19,00	Suicide	12,000
D. Heart Disease	36,000	Lung Cancer	37,000
E. Homicide	9,200	Flood	100
F. Diabetes	19,000	Syphilis	200
G. Asthma	900	Botulism	1
H. Lightning	52	Poisoning/Vitamins	.5
I. Homicide	9,200	Tuberculosis	1,800
J. All Accidents	55,000	Stomach Cancer	46,600

Statistical Use and Abuse

There are three kinds of lies: lies, damned lies, and statistics.

—Disraeli (1804–1881)

When we want to find out something about a group of people, it is often impossible or inconvenient to ask everyone in the group. Suppose you want to know if people who donate blood to the Red Cross are, in general, kind and generous people. Because you cannot examine everyone who donates blood to determine how kind and considerate she or he is, you would examine a portion of the population, which is called a sample. A number calculated on a sample of people is called a statistic. (“Statistics” is also the branch of mathematics that utilizes probability theory to make decisions about populations.)

Statistics are found everywhere, from baseball earned run averages to the number of war casualties. Many people are rightfully suspicious of statistics. A small book by Huff (1954) humorously illustrates many of the possible pitfalls of statistics. The book is entitled How to Lie With Statistics. In it, he rhymes the following message: “Like the ‘little dash of powder, little pot of paint,’ statistics are making many an important fact look like what she ain’t” (p. 9). But, the real message is that you will not be bamboozled with statistics if you understand them.

On the Average

What does it mean to say that the average American family has 2.1 children? This number was computed by finding a sample of American families, adding up the number of children they have, and dividing by the total number of families in the sample. This number could provide an accurate picture of American families, as most may have about two children, with some having more and others less, or it could be very misleading. It is possible that half of the families had no children and half had four or more children, thus misleading the reader into believing that most families had “about” two children, when in fact none did. This is like the man who had his head in the oven and feet in the freezer and reports that, on the average, he is quite comfortable. It is also possible that the sample used to calculate this statistic was not representative of the population—in this case, all American families. If the sample consisted of college students or residents in Manhattan, the number obtained would be too low. On the other hand, if the sample was taken in rural farm areas, the number obtained may be too high. When samples are not representative of the population, they are called biased samples. The statistics calculated on biased samples will not yield accurate information about the population.

Averages can also be misleading since there are three different kinds of averages. Consider Mrs. Wang’s five children. The oldest is a successful corporate executive. She earns $500,000 a year. The second is a teacher who earns $25,000 a year. Child 3 is a waiter who earns $15,000 a year. The other two are starving artists, each earning $5,000 a year. If Mrs. Wang wants to brag about how well her children turned out, she could compute an arithmetic average, which is called the mean. The mean is what most people have in mind when they think about averages. It is the sum of all of the values divided by the total number of values. The mean income for Mrs. Wang’s children is $550,000/5 = $110,000. Certainly, anyone who is told this figure would conclude that Mrs. Wang has very successful and wealthy children.

The reason that the mean income for Mrs. Wang’s children was so high is because there is one extreme score that inflated this type of average. Averages are also called measures of central tendency. A second kind of measure of central tendency is the median. It is not affected by a few extreme scores. To compute the median, the values are lined up in ascending or descending order. The middle value is the median. For Mrs. Wang’s children, this would be:

$5,000; $5,000; $15,000; $25,000; $500,000.

The middle value, or median, is the third value of $15,000. Thus, she could also claim that her children earn, on the average, $15,000. (When there is an even number of values, the median is equal to the mean of the middle two values.) Mrs. Wang can honestly claim that her children earn, on the average, $110,000 or $15,000. The point of this discussion is that you should be wary of average figures. To understand them, you need to know whether the average is a mean or median as well as something about the variability of the data and the “shape” of the distribution (the way the numbers “stack up”).

Precision

Suppose I tell you that a scientific survey was conducted on the length of workdays for office workers. Furthermore, this study found that the mean workday is 8.167 hours long. Does this sound impressive and scientific? What if I told you that most office workers work about 8 hours a day? Most of you would say, “I know that. Why did they bother?” The point is that we are often impressed with precise statistics, even when the precision is unwarranted. A more humorous example of over-precision comes from one of America’s most famous authors, Mark Twain. He once reported that the Mississippi River was 100 million and 3 years old. It seems that three years earlier, he learned that it was 100 million years old.

Significant Differences

If you wanted to know the mean height of all women, you could select a sample of one hundred women, measure their height, and compute the mean. Suppose you took another sample of one hundred women and computed their mean height. Would you expect the means of these two samples to be exactly the same? No, of course not, since there could be expected to be small differences or fluctuations between these values. Each value was computed on different women, and each will yield a slightly different mean value.

If someone measured a sample of women who belong to sororities and found that their mean height is 5’5” and then measured women who do not belong to sororities and found their mean height to be 5’4 1/2”, would you conclude that sorority sisters are taller than nonsorority women? I hope not, because small differences between groups can be expected to occur just by chance. There are statistical procedures to determine if a difference computed on two or more samples is likely to have happened by chance. If it is very unlikely to be a chance occurrence, it is called a significant difference.

The question of whether a change is meaningful also applies to populations as well. If your college enrollment went from 15,862 to 15,879, would the administrators be justified in concluding that the increase in enrollment is meaningful? The answer to this question depends on many other variables. If the enrollment figure has been edging up slowly every year for the last five years, then these figures may represent a slight, but steady, trend. On the other hand, the relatively small increase could be due to chance fluctuations and may not represent a meaningful trend. Because of chance factors, it could just as easily have gone down. Similarly, a change in the unemployment rate from 10.0% to 9.9% may be nothing more than random fluctuation, or it may be signaling the end of an economic quagmire. You can expect that Democrats and Republicans will interpret these figures differently depending on who is in office at the time.

Extrapolation

Extrapolation occurs when a value is estimated by extending some known values. If the number of psychology majors over the last five years at Podunck University was approximately 150, 175, 200, 225, and 250, respectively, then most people would feel comfortable about the prediction that the number of psychology majors next year will be approximately 275.

From www.CartoonStock.com. Used with permisssion.

Extrapolation can be wrong, and sometimes even ridiculous. For example, suppose we were to examine the drop in the size of American families between 1900 and 1950. By extrapolation, we would predict that the average family size will soon be zero, and then become a negative number. This is, of course, an inconceivable idea! This is like saying that if the times for the 100-meter dash keep decreasing, eventually someone will run it in zero seconds and then in negative time.

Chapter Summary

• Because few things are known with certainty, probability plays a crucial role in many aspects of our lives.

• Probability is defined as the number of ways a particular outcome (what we call a success) can occur divided by the number of possible outcomes (when all outcomes are equally likely). It is also used to indicate degrees of belief in the likelihood of events when the objective probabilities are unknown. People are often inaccurate when they estimate the likelihood of future events.

• In general, people tend to be more confident about uncertain events than the objective probability values allow. We underestimate the role of chance when thinking about many events and invent our own explanations for random outcomes.

• Mathematically equivalent changes in the way probability information is presented can lead to dramatic changes in the way it is interpreted.

• Tree diagrams can be used to compute probabilities when there are multiple events (e.g., two or more flips of a coin). When the events are independent, the probability of any combination of outcomes can be determined by multiplying the probability values along the tree “branches.”

• Expected values can be computed that will take into account the probabilities and values associated with a loss and a win in betting situations. One real-life problem with expected values is that people often misjudge the value of an outcome (e.g., how much they will enjoy or dislike something).

• Subjective probabilities are our personal estimates of how likely events with unknown frequencies will occur. These values are distorted systematically when people believe that they have some control over probabilistic events.

• Most people fail to consider the cumulative nature of the likelihood of risky events.

• People judge events that are dramatic and more publicized to be more likely than events that are less dramatic or less well known. In general, people overestimate frequent events and underestimate infrequent ones.

• There is a tendency to ignore base-rate information, especially when making predictions that involve combining information.

• Few people realize that if a person scores extremely high or low on one measure, she or he will tend to score closer to average on the second measure. Regression to the mean is not an intuitive concept, but it is relevant in many settings.

• There are two measures of central tendency that are frequently used—the mean and median. Each is computed with a different mathematical formula.

• There are several systematic biases that operate when most people assess risks. These include downgrading the probability of voluntary risks and those over which we have some perceived control, and overestimating risks that are artificial, memorial, and unobservable.

• Many people erroneously believe that statistics that are expressed in precise numbers (e.g., many decimal places) are highly credible.

• Extrapolation occurs when a value is estimated by extending a trend from known values. Depending on what is being extrapolated and the quality of the data, our best bets for future outcomes can range from fairly accurate to wildly unbelievable.

The following skills for understanding likelihood and uncertainty were presented in this chapter. Review each skill and be sure that you understand how and when to use each one.

• computing expected values in situations with known probabilities

• recognizing when regression to the mean is operating and adjusting predictions to take this phenomenon into account

• using the “and rule” to avoid conjunction errors

• using the “or rule” to calculate cumulative probabilities

• recognizing and avoiding Gambler’s Fallacy

• utilizing base rates when making predictions

• using tree diagrams as a decision-making aid in probabilistic situations

• adjusting risk assessments to account for the cumulative nature of probabilistic events

• understanding the differences between mean and median

• avoiding overconfidence in uncertain situations

• understanding the limits of extrapolation

• using probability judgments to improve decision making

• considering indicators like historical data, risks associated with different parts of a decision, and analogies when dealing with unknown risks.

Terms to Know

Check your understanding of the concepts presented in this chapter by reviewing their definitions. If you find that you are having difficulty with any term, be sure to reread the section where it is discussed.

Probability. The number of ways a particular event can occur divided by the number of possible outcomes (when all outcomes are equally likely and the events are independent). It is a measure of how often we expect an event to occur in the long run. Probability is also used to express degrees of belief in the likelihood of a future event when there is no objective information about its occurrence.

In the Long Run. Refers to the need for numerous trials in order to derive accurate estimates of the proportion of outcomes that will be a “success.” The laws of probability apply to outcomes when there are many trials and not to the outcome on any single trial.

Odds. A mathematical method for indicating probability that is commonly used in sporting events.

Laws of Chance (or Probability). The ability to predict the number or percentage of trials on which a particular outcome will occur.

Overconfidence Phenomenon. The tendency for people to be more confident in their judgments of probability than the objective probability values allow.

Multiple Outcomes. Refers to the probability of an event occurring in two or more trials, such as getting two heads in two flips of a coin.

Tree Diagrams. Branching diagrams that may be used to compute probabilities by considering all possible outcomes in a sequence of events.

Independent Events. Two or more events are independent when the occurrence of one event does not affect the occurrence of the other events.

Conjunctive Error. Mistaken belief that the co-occurrence of two or more events is more likely than the occurrence of one of the events alone.

Cumulative Probabilities. The probability of an event occurring over many trials.

Expected Value. The amount of money you would expect to win in the long run in a betting situation. The mathematical formula for determining expected values is the probability of winning times the value of winning plus the probability of losing times the value of losing.

Subjective Probability. Personal estimates of the probability or likelihood of uncertain events.

Objective Probability. Mathematically determined statements about the likelihood of events with known frequencies.

Affective Forecasting. People tend to be poor at predicting (forecasting) how they will feel in the future if a particular event occurs.

Base Rate. Initial or a priori probability that an event will occur.

Base-Rate Neglect. Pervasive bias to ignore or underestimate the effect of initial probabilities (base rates) and to emphasize secondary probability values when deciding on the likelihood of an outcome.

Gambler’s Fallacy. The mistaken belief that chance events are self-correcting. Many people erroneously believe that if a random event has not occurred recently, it becomes more likely.

Relative Frequency. How often an event occurs relative to the size of the population of events at the time of its occurrence.

False Positive. A false positive outcome occurs when a test incorrectly predicts an outcome, such as when a test erroneously identifies someone with a disease or other characteristic.

Regression Toward the Mean. In general, when someone scores extremely high or low on some measure, she or he will tend to score closer toward the mean (average) in a second measurement.

Sample. A subset of a population that is studied in order to make inferences about the population.

Statistic. A number that has been calculated to describe a sample. (In its plural form, it is the branch of mathematics that is concerned with probabilities and mathematical characteristics of distributions of numbers.)

Biased Sample. A sample that is not representative of the population from which it was drawn, so you cannot use information from that sample to make inferences about the population.

Measures of Central Tendency. Numbers calculated on samples or populations that give a single number summary of all of the values. Two measures of central tendency are the mean and median.

Mean. A measure of central tendency that is calculated by taking the sum of all the values divided by the total number of values. Also called the average.

Median. A measure of central tendency that is calculated by finding the middle value in a set of scores.

Sample Size. The number of people selected for a study. Samples need to be large enough (actual values are not given here) to make a valid inference about the population.

Significant Difference. A difference between two groups or observations that is so large that it probably did not occur by chance. Statistical tests are used to determine if a difference between samples is statistically significantly different.

Extrapolation. The estimation of a value from a trend suggested by known values.