• Knowing how to use numbers to measure likelihood
• Analyzing combinations of events
• Understanding how different events can affect each other
There are many aspects of the world that can be measured with numbers. This is what makes mathematics is so endlessly fascinating! But this is not limited to things that we can weigh or measure. Some applications of numbers are subtler, and more indirect. One important area is the study of probability, where we use numbers to represent the likelihood of certain events taking place.
When we talk about an event being “likely” to happen, or “certain” to happen, we are using the language of probability. In the study of probability, we assign a number to this likelihood to quantify the chance of the event happening. We use only the numbers between 0 and 1: an impossible event has probability 0, while a certainty has a probability of 1. Everything else falls somewhere in between. For example, if I toss a coin, then the probability of it landing on heads is 
(one chance in two), so long as the coin is fair.
Biased coins will have other probabilities. To take an extreme example, a double-headed coin (a coin with a head on each side) will have probability 1 of landing on heads (it is certain to happen). For the rest of this chapter we will make a standing assumption that all the coins (and dice and decks of cards) we meet are fair.
At one end of the scale, unlikely events have very small probabilities, meaning numbers close to 0. The chance of your ticket winning the UK National Lottery or the Washington State Lottery are each 
(around 0.00000007). Events which are completely impossible have a probability of 0. (As lotteries like to advertise: if you don’t have a ticket, your chance of winning is absolutely nil!)
At the other end of the scale, very likely events have high probabilities, meaning numbers close to 1. The probability that the sun will rise tomorrow is very close indeed to 1, something like 0.9999…999. (I wouldn’t want to guess the number of 9s, but in the 18th century the naturalist George-Louis Leclerc made a serious attempt to estimate it!)
If I am asked to give a rough and ready estimate of how likely it is to rain tomorrow, I might start by reasoning that, at this time of year, it typically rains in my town around one day in two. This would put the chance of it raining tomorrow at around 0.5. If it has been raining across the entire region for several days and shows no signs of clearing up, I might increase that estimate to, say, 0.8.
TRY ESTIMATING SOME PROBABILITIES IN QUIZ 1.
It is all very well estimating the probabilities of events according to how likely they seem. But how can we work out exact answers? One basic technique amounts to counting the outcomes of an experiment.
When we roll a standard die, there are six possible outcomes (1, 2, 3, 4, 5, 6). Suppose I want to know the probability of rolling a 5. Just one of the six outcomes counts as a “success,” which gives us our answer: a probability of 
. The rule here is that the total number of possible outcomes goes on the bottom of the fraction, and the number of “successful” outcomes goes on the top:
This is the basic idea. But as usual there is some fine print to take account of! If we go back to the question of whether or not the sun will rise tomorrow, then there are two possible outcomes: either it will or it won’t. Of these, just one (sunrise), is classed as a “success.” So, according the rule above, the answer should be .
The trouble with this is obvious! It is just nonsense. Sunrise is a near certainty, and so should have a probability very close to 1.
So what has gone wrong? Well, when counting up successes and outcomes, there is an additional rule: that all the possible outcomes must be equally likely. This is what fails in the case of the sun. So, for the formula to work, the dice and coins used must be fair.
IT’S TIME FOR QUIZ 2.
What is the point of assigning numbers to the probabilities of events? It is not just because mathematicians are fixated with measuring everything numerically. One benefit is that different ways of combining events correspond very neatly to various arithmetical tricks with their probabilities. There are two principal cases of this, which are described by the two English words “and” and “or.”
Let us take “and” first. Suppose I roll a die and flip a coin. What is the probability that I will roll a 6 and flip a head? We know that the probability of rolling a 6 is , and the probability of getting a head is . How do we mix these numbers, to get the probability for the combined event of a head and a 6. The answer is to multiply. So the probability we want is 
, which comes out as 
.
The general rule here is that “and” in a combined event means “multiply” the probabilities. But we cannot just apply this rule blindly; again there is some fine print to take into account. What, for example, is the probability that, when I flip a coin once, I get both a head and a tail? The answer should be 0, since that event is completely impossible. But if we apply the rule above, without thinking about what it means, we get an answer of 
.
What is the caveat we need to eliminate nonsense like this, and leave us with a rule that makes sense? The answer is that the two events whose probabilities we are multiplying must not affect each other. In technical terms, they must be independent. The first example passes this test: when I roll a die and toss a coin, whether or not I get a head has no impact on whether or not I get a 6. But, in the second example, with just one coin, whether or not I get a head makes a huge difference to the likelihood of my getting a tail. (In fact the one determines the other entirely.)
IT’S TIME TO HAVE A GO AT QUIZZES 3 AND 4.
So we can express the rule more accurately: when two events are independent, “and” means “multiply.”
Let’s move on to the other principal way that two events can be combined: “or.” When I roll a die, I might be interested in the probability of my getting a 5 or a 6. Here we can move directly to the method of counting up successes and outcomes, which will quickly give us the answer: 
(which can be simplified to 
: see Fractions). But it is useful to think about how this answer is related to the individual probabilities of the two separate events: getting a 5 or getting a 6. Each of these has probability . The probability of the combined event, a 5 or a 6, comes from adding these two together.
So the general rule is, when finding the probability of a combined event: “or” means “add.”
Let’s have another example: Suppose I pick a card from a deck. The probability of getting a heart is 
. The probability of getting the queen of spades is 
. So what is the probability of getting a heart or the queen of spades? Well we can apply the simple rule—“or” means “add”—to get an answer of 
which comes out as 
once the fractions have been added and simplifed. (Try working that through yourself!)
As ever, though, caution is needed because this rule also comes with some fine print. Here is why: Suppose I flip two coins. What is the probability that I will get two heads? If I unthinkingly apply this rule, I would reason as follows: The probability that I get a head on the first coin is . The probability that I get a head on the second coin is too. “Or” means “add,” so the probability that I get a head on the first coin or the second coin is 
. This suggests that it is a certainty. But of course this is nonsense: it is perfectly possible that I will get two tails.
The fine print in this case is that you can only add together the probabilities of two events when they cannot both occur. When I roll a die, I cannot get both a 5 and a 6. So it is safe to add together those probabilities. But I can get heads on two coins, so I am not allowed just to add together those probabilities. In the jargon, the two events must be mutually exclusive. This means that if one happens, then the other doesn’t.
If I roll one die, the two outcomes (a 5 and a 6) are mutually exclusive. But if I flip two coins, the two outcomes (a head and a head) are not mutually exclusive.
HAVE A GO AT QUIZ 5
Now we can express the rule more accurately: when two events are mutually exclusive, “or” means “add.”
Sum up Whenever you think something is “impossible,” “unlikely” or “certain,” you are using the language of probability. It has techniques to assess the likelihood of different events happening—a valuable prize in this uncertain world!
1 Guess approximate probabilities for the following events (answers may vary from person to person!).
a The next person you meet will be male.
b An asteroid will hit your house tomorrow.
c If you turn on the TV, the first person you see will be wearing glasses.
d If you pick a word on this page at random, it will have an “e” in it.
e Your favorite sports team will win their next match.
2 Calculate these probabilities by adding up the total number of successes and outcomes.
a You pick a playing card from an ordinary deck. What is the chance of getting an ace?
b You roll an ordinary die. What is the chance of getting an even number?
c You roll a 12-sided die. What is the chance of getting an 8 or higher?
d You pick a card from an ordinary deck. What is the chance of getting a spade?
e You roll a 20-sided die. What is the chance of getting a prime number?
3 Are these pairs of events independent?
a You toss a 10c coin and a 25c coin and get a head on the 10c and a tail on the 25c coin.
b You roll a die and pick a card from a deck. You get an even number on the die and a king.
c You pick a card from a deck, replace it, shuffle, and pick another card. Both times you get an ace.
d You pick a card from a deck, don’t replace it, and then pick another. Both times you get an ace.
e You pick a single card from a deck and get a black card and an ace.
4 In quiz 3 above, where the pairs of events are independent, calculate the combined probability of both occurring.
5 Are these pairs of events mutually exclusive? Where the answer is yes, calculate the probability of one or the other happening.
a You pick a card from a deck and get the queen of spades or a heart.
b You roll a die and get an odd number or a 6.
c You toss a 10c coin and a 25c coin and get a head on the 10c coin or a head on the 25c coin.
d You roll two ordinary dice, and their total is 2 or their total is 12.
e You pick a card from the deck, replace it, shuffle and pick again. You get the ace of spades twice, or the queen of hearts twice.