8

No Such Thing as Coincidence?

If mainstream science is right to reject the idea that paranormal forces exist, how are we to account for the fact that a sizable proportion of the general population claim to have had direct personal experiences of such forces? As we have seen in previous chapters, many of the more extreme claims can be accounted for in terms of unusual activity within the brain and the fact that hallucinatory experiences and false or distorted memories are much more commonplace than most people realize.

For many people, however, their general belief in the paranormal is often reinforced by more frequent, less spectacular, ostensibly paranormal events, such as knowing who is calling before picking up the phone, dreams that appear to come true, and acting on hunches that turn out to be correct. This chapter and the following one will consider the ways in which many of these experiences can be explained in terms of cognitive biases—that is, underlying biases in how we process information from the world around us that may lead us to believe that we have had a paranormal experience when, in fact, plausible nonparanormal explanations exist.

I first reviewed the research literature on this topic back in 1992. The tasks I set myself then were, “firstly, to demonstrate that certain situations are likely to be misinterpreted as described above and, secondly, that believers (sheep) may be more prone to such misinterpretation than disbelievers (goats).” Back then, there was not a great deal of relevant research to review. Based on what there was, I concluded, “It is clear that there is plenty of evidence to show that cognitive biases do exist which would lead to people misinterpreting certain situations as necessarily involving psi when in fact perfectly acceptable nonparanormal explanations are available.” I went on to say, “The question of whether believers are more prone to such biases than goats cannot be answered with certainty, but the limited evidence available suggests that this is a real possibility.”1

Fifteen years later, in collaboration with Krissy Wilson, I published an updated review of research in this area.2 By this point, considerably more research had been carried out. Some of the cognitive biases identified in the early research did indeed appear to be stronger in believers than skeptics, but findings were mixed with respect to others. Now, some three decades after my initial review, there have been too many additional studies to comprehensively review in two chapters, but the overall picture remains the same.3 This chapter will present an overview of research into the psychology of coincidences, and the chapter 9 will consider other cognitive biases of potential relevance.

What Are the Chances?

As I was preparing to write this chapter, my sister-in-law, Jane Hatzimasouras, returned from a holiday on the Greek island of Spetses with a rather remarkable true story to tell. One day she was passing the time in a café with friends. One of her party returned from a swim with a ring that he had found on the seabed near a rock about 40 yards away from the main shoreline. A name and date were engraved on the inside of the ring: Athena Karagiannis and November 17, 1979.4 It is traditional in Greece when a couple marry for the groom to have a ring engraved with his wife’s name, along with the date of their wedding, and vice versa for the bride.

As the group excitedly mused on how the ring had come to be in its watery hiding place, Jane could not help but notice a woman who was not one of her group listening in intently. Eventually, this woman approached my sister-in-law’s party and said, “I hope you don’t mind me interrupting, but I could not help but overhear you discussing the ring that you have found. You see, my surname is Karagiannis and my mother’s name is Athena—and about forty years ago, my dad lost his ring while swimming in the sea!”

The group then waited in anticipation as the stranger called her mother to check on the date of her mother’s wedding day, only to be disappointed to learn that it had actually been March 8, 1980. It appeared that the ring was not the one that had been lost four decades earlier after all. But then the woman called her father to tell him what had happened, and he explained that he had had the rings engraved not with the wedding date but their engagement date—and that date was indeed November 17, 1979!

What were the chances that Mr. Karagiannis’s daughter would just happen to be in the right place at the right time to overhear a conversation about a found ring that happened to be the very same ring that he had lost some four decades earlier? I think we can all agree that, even if we cannot work out the exact probability of this co-occurrence of events, it must be astronomically low. For some people, the probability would be so low that they would be strongly inclined to believe that some unseen force—perhaps fate?—had a role to play. Surely, something more than blind chance must have been at work here?

There is no denying that coincidences such as these have a definite emotional impact. They are the sort of thing that we cannot wait to tell our friends about, often beginning our anecdote with the words, “You’ll never believe what happened to me!” Sometimes they end up being reported in newspapers and on TV and may be featured in books and academic articles. Here are a few of my personal favorites.

David Hand includes a lost-and-found wedding ring example of his own in his wonderful book The Improbability Principle.5 In 1995, in Sweden, Lena Påhlsson lost her wedding ring, only to find it encircling a carrot she pulled up in the garden in 2011. However, Hand begins his book with an even more amazing coincidence:

In the summer of 1972, the actor Anthony Hopkins was signed to play a leading role in a film based on George Feifer’s novel The Girl from Petrovka, so he travelled to London to buy a copy of the book. Unfortunately, none of the main London bookshops had a copy. Then, on his way home, waiting for an underground train at Leicester Square station, he saw a discarded book lying on the seat next to him. It was a copy of The Girl from Petrovka.

As if that was not coincidence enough, more was to follow. Later, when he had a chance to meet the author, Hopkins told him about this strange occurrence. Feifer was interested. He said that in November 1971 he had lent a friend a copy of the book—a uniquely annotated copy in which he had made notes on turning the British English into American English (‘labour’ to ‘labor’ and so on) for the publication of an American version—but his friend had lost the copy in Bayswater, London. A quick check of the annotations in the copy Hopkins had found showed that it was the very same copy that Feifer’s friend had mislaid.

Journalist and broadcaster Simon Hoggart recounted the following tale in his book, coauthored with Mike Hutchinson, Bizarre Beliefs:

In 1994, Simon Hoggart’s wife was traveling with a friend on a tube train. They were to visit an exhibition in Central London. She was talking about the actor, Richard E. Grant, who she did not know, but who lived in her neighbourhood. The friend said she could not place him; what did he look like? At that moment, Richard E. Grant got on to the train, and sat opposite them. He was holding a ticket for the same exhibition they were going to.6

Perhaps there is something about tube stations that facilitates the occurrence of amazing coincidences?

Research shows that we tend to be more impressed by coincidences that involve us personally than by the same coincidences happening to other people, so I will risk you being much less impressed than I was with the following coincidence that I experienced as a teenager.7 At the time, I embraced a lot of New Age beliefs. Eager to learn more, I was reading the book The Morning of Magicians by Louis Pauwels and Jacques Bergier, a book crammed to breaking point with what I would now view as New Age nonsense, pseudoscience, and pseudohistory.8 At the time, of course, I lapped it up. As I read it, I had David Bowie’s Hunky Dory album playing in the background, specifically the track “Quicksand.” I had listened to the album many times but did not understand all of the references in Bowie’s poetic lyrics. Then, at the very moment that Bowie sang the line, “I’m closer to the Golden Dawn, immersed in Crowley’s uniform of imagery,” I read the words in Pauwels and Berger’s book about the occult Golden Dawn society, one member of which was the infamous Aleister Crowley. Up until that point, I had never really heard of either. Needless to say, I saw this event as having tremendous mystical significance.

As with all claims of miraculous events, it is wise to be a little cautious in accepting all claims of amazing coincidences at face value. For example, Martin Plimmer and Brian King include the following account in their book Beyond Coincidence:

In June 2001, ten-year-old Laura Buxton of Burton in Staffordshire was at a party where she wrote her name and address on a luggage label, attached it to a helium balloon and released it into a clear blue sky.

The balloon floated 140 miles until finally coming to rest in the garden of another Laura Buxton, aged ten, in Pewsey, Wiltshire.9

A lovely, heartwarming little story, the only problem being that it isn’t quite true. According to Snopes, checking back to the original story in the Swindon Advertiser reveals that the balloon did not land in (the second) Laura Buxton’s garden but was found by farmer Andy Rivers in a field while out checking his cows.10 Mr. Rivers knew his neighbors, the Buxtons, had a daughter called Laura so he “returned” the balloon to them. This was not, of course, the Laura Buxton who had launched the balloon (also, for fellow pedants out there, it turns out that the second Laura was nine, not ten, years old). This is still quite an interesting coincidence, but the balloon being found by someone who knew of a Laura Buxton is nowhere near as unlikely as the balloon being found by someone named Laura Buxton.

Despite the occasional exaggerated claim, there are numerous well-documented accounts of amazing coincidences. For example, people really do get hit by lightning more than once. In fact, Virginia park ranger Roy Sullivan was hit by lightning at least seven times (he claims he was also struck by lightning as a child, but we only have his word for it).11 There are numerous examples of people getting very, very rich by winning lottery jackpots more than once. In June 1980, Maureen Wilcox found she had the winning numbers in both the Rhode Island Lottery and the Massachusetts Lottery.12 Unfortunately for her, however, she had the winning numbers for the Rhode Island Lottery on the Massachusetts ticket and vice versa, so she did not win a cent—but, hey, it’s still a pretty amazing coincidence!

Given our fascination with coincidences, it is not surprising that they have attracted the attention of both mathematicians and psychologists.13 One of the reasons that we are more impressed with coincidences than perhaps we should be is the fact that we are all extremely poor intuitive statisticians.14 In other words, in everyday life we frequently have to make decisions based on real-life probabilities, and we are simply not very good at doing so. Our poor intuitive understanding of probability can manifest itself in a variety of ways. One example is our frequent failure to appreciate what mathematicians refer to as the law of truly large numbers. This law states that with a large enough number of opportunities for an event to occur, even extremely unlikely events become probable.

The most obvious and familiar example of the law of truly large numbers in action is national lotteries. For example, the chances of winning the national lottery in the United Kingdom at the time of writing are around one in forty-five million—and yet we are not surprised when people regularly win the jackpot because, in this context, we fully appreciate that so many tickets are bought that it is indeed quite likely that at least one of them may match all six randomly drawn numbers (between 1 and 59).

In other contexts, the operation of the law does not appear to spring to mind as an adequate explanation for many people, such as when the same person wins the jackpot more than once. When newspapers report double winners, the probabilities are often reported in a way that, although technically correct, are not actually relevant to the question that should really be asked. Implicitly, the question that is usually being asked is “What is the probability that person X would win the jackpot in this particular lottery twice if they only ever bought two tickets?” It is immediately obvious with the emphasis added here that X may well have bought more than one ticket on each of the two days they bought their winning tickets. That would clearly increase the odds of them winning on each of those days. In fact, they probably bought multiple tickets on lots of other days as well, further improving their chances of winning the jackpot. But we can go even further. If we consider the fact that there would be a “truly large” number of people behaving in the same way for that particular lottery, it becomes even more likely that we will get a double winner. The final level comes when we consider that there are numerous national, international, and state lotteries in operation around the world. It becomes almost inevitable that multiple winners will crop up—and end up being reported in the media.

Consider the law of truly large numbers as it applies to my David Bowie/Golden Dawn example. I have no idea how many people had bought the Hunky Dory album by the time I was listening to it on that occasion, but I know it would be a “truly large number.” A lot of those people would no doubt, like me, be playing it repeatedly. I suspect a smaller, but still very large, number of people read the cult classic The Morning of Magicians. Some of those people would, like teenage me, be in the habit of playing records while reading. It suddenly seems not quite so surprising that one of those people experienced the coincidence that I reported—it just happened to be me.

Sometimes the law of truly large numbers can operate in ways that are much less obvious than in the case of national lotteries. One nice example is the so-called Birthday Problem. The problem is usually stated as follows: How many randomly selected people would you need to have at a party to have a fifty-fifty chance that two of them share the same birthday (ignoring year)?15 We will ignore leap years to keep things simple. If you have not come across this problem before, give it some thought before I reveal the correct answer. If you are very mathematically gifted, it is, of course, possible to work out the exact answer, but that is not a trivially easy task even then. We are more interested here in what you intuitively feel is approximately the right answer.

• *** Short (and probably imaginary) musical interlude ***

Okay, here is the answer, presented in the form of an endnote (to minimize the chances of you just happening to glance at it accidentally while thinking about the problem).16 That answer strikes most people as being far too low. It may be that, in some cases, people have misunderstood the problem as being “how many people would you need to have at a party to have a fifty-fifty chance that they share your birthday?” But even when it is made very clear that the problem is asking about the probability of any pair of people having the same birthday, people typically greatly overestimate the number required.

If you are still not convinced that the answer I presented is correct, you might like to consult, say, David Hand’s book to see how the answer is calculated.17 Or, if you are able to write simple computer programs, you could write a bit of code to generate random numbers between 1 and 365 (to correspond to each day of the year) repeatedly until a number is generated that has already been produced. You will see that the strings of numbers generated are less than or equal to twenty-three numbers long about half of the time. Or you may be convinced by visualizing someone throwing darts at random at a dart board divided into 365 sectors. Of the darts that actually hit the board, the first one could land in any sector and the second one would only have one chance in 365 of landing in the same sector. The third dart would have two chances in 365 of landing in the same sector as one of the first two—still very unlikely—but as more and more darts are randomly thrown, the chances of such an outcome would be improving all the time. By the time, twenty or more darts have been thrown, we might not be too surprised to see a dart land in an occupied sector.

What we typically fail to intuitively take into account when we first think about the Birthday Problem is the fact that the number of possible pairings increases exponentially as we add in each new person (or dart). When there are twenty-three people in the room, there are 253 possible pairings, so having at least one pair with the same birthday is not at all unlikely.

Another factor that has been identified as potentially resulting in what appear to be astronomically unlikely coincidences is that of hidden causes. A nice example of this, discussed by James Alcock, is the “perfect” bridge deal, in which each of four players is dealt thirteen cards of the same suit from a shuffled deck.18 Now, it should be noted here that such an outcome is no more or less likely than any other prespecified bridge hand, but it is certainly more noticeable—and, like any other prespecified hand, it is exceedingly unlikely. There are, in fact, approximately 2.23 x 10²⁷ (that’s 223 followed by twenty-five zeros—indubitably, a truly large number!) possible distributions. To bring home just how mind-blowingly huge that number is, Alcock provides the following quotation from the McWhirter brothers: “Such an event having once occurred, it should not logically recur, even if the entire world population made up in fours and played 120 hands of bridge a day, for another 2,000,000,000,000 years.”19

Given the extreme improbability of such bridge deals, it is therefore puzzling that cases of perfect bridge deals are reported every year. You may be tempted to concur with the McWhirters that such reports must due to either “rigged shuffling or hoaxing.” Both possibilities would be good examples of hidden causes, but Alcock notes an even subtler possibility. New decks of cards are almost always organized in ascending order by suit. This means that if such a deck were subjected to two “perfect” shuffles (in which the deck was divided into two equal halves, which were then perfectly interleaved), the deal would always be a perfect bridge deal. It is therefore not that unlikely that this would indeed happen from time to time.

The more exact the coincidental match between the elements involved, the more impressed we are by the coincidence—but we are often almost as impressed by an almost exact match. For example, when giving public talks sometimes I illustrate the probabilities involved in the Birthday Problem by asking audience members to shout out, in turn, their birthdays until we reach someone whose birthday has already been called out.20 As you will now expect, this point is reached about half the time with twenty-three or fewer members of the audience contributing. But people will often spontaneously comment when a previous audience member has shouted out a birthday either a day before or a day after their own. As another example, we would no doubt be very impressed when, as can happen, someone wins the jackpot twice in the same lottery—but we’d probably be almost as impressed if they won the jackpot twice in different lotteries or won the jackpot and a smaller, but still substantial prize, in the same lottery. Needless to say, our willingness to count close matches as being almost as impressive as exact matches will greatly increase the chances of such matches occurring.

Coincidence and Ostensibly Paranormal Experiences

Why are coincidences relevant to ostensibly paranormal experiences? The answer is pretty obvious. In many cases, the most obvious nonparanormal counter-explanation is simple coincidence. At first sight, “mere coincidence” appears to be a very poor explanation for such events. Put yourself in the position of someone who has not heard from or even thought about a long-lost friend for several years who, just as they are thinking of that person, gets a call from them. Could that really be just a coincidence?

Not only does dismissing the event in that way feel wrong on an emotional level, it does not initially appear to provide an adequate explanation on an intellectual level either. After all, if that really was the first time that you thought about your friend in many years, surely the chances were literally several million to one against them just happening to call you at that precise moment? But a moment’s reflection will reveal that coincidence may well be a more plausible explanation than any kind of spooky psychic link. Just consider how many people you think about in a single day. Most of the time when you think about them, either the phone does not ring or the phone rings and it is someone else calling. But, occasionally, simply based on the laws of probability, the person calling will be the person you were thinking about. In fact, it would be really spooky if this never happened.

Similarly, in the case of dreams that appear to come true, we tend not to think about the fact that pretty much everyone dreams pretty much every night. It would be amazing if no one ever had a dream that corresponded, purely by chance, to some unexpected event in real life sometime later. As we have seen, according to the law of truly large numbers, even very unlikely coincidences are likely to happen given enough opportunities.

American mathematician John Allen Paulos provides a nice illustration of how the law of truly large numbers applies to ostensibly precognitive dreams.21 Let us suppose that you would only be willing to label a dream as “potentially precognitive” if details within the dream matched future events in real life with an estimated probability of occurring by chance of only one in 10,000. Surely, all of us would be pretty impressed if we were the one to have such a dream?

That means that on any particular night the chances are very high indeed that we would not have such a dream: 9,999 times out of 10,000, in fact. The chances of us not having such a dream over a period of a whole year are still very high. The actual probability can be calculated by multiplying 0.9999 by itself 365 times. That is about 0.964, meaning that 96.4 percent of people who dream every night will not have a “potentially precognitive” dream over a one-year period. But that means that 3.6 percent of those people will have such a dream—and that translates into millions of people worldwide.

The question that immediately springs to mind in such circumstances is “What are the chances that that particular coincidence would occur to me?” But to decide whether the event involves anything that cannot be explained purely in terms of probability theory, the question we should really be asking is “What are the chances that anyone at any time would experience any coincidence as unlikely as that?” That way, we can immediately see that anyone who experienced such an unlikely coincidence was bound to be blown away by it—it just happened to be you!

Another overlooked factor in such situations is that there is often more at work than pure chance even though we may not be aware of it. One such hidden factor in our hypothetical example of the phone call from the long-lost friend would be anything in the real world that would cause you both to start thinking of each other after a long period of not doing so. For example, suppose that you were both fans of a particular musician during your college days and frequently went to their concerts. Over the years, your musical tastes may have moved on, but you and your friend both happened to catch a minor news story about this (now aging) musician. That got you both reflecting on those carefree days at college and the many good times you had together. By the time your friend decides to track you down and give you a call, you might have both forgotten what prompted you into this train of thought in the first place—and you would be amazed that your long-lost friend just happened to call you “out of the blue” just as you were thinking about them.

The principle of “near enough is good enough” crops up in numerous paranormal contexts. For example, suppose it was not our friend who called us out of the blue just as we were thinking about him but his mother, father, brother, or sister, or a mutual friend, and so on. We would still be very impressed by the coincidence. Suppose we had a dream of being involved in a serious car accident as we were driving to work, and the very next day we had a minor bump driving to the shops. Again, we may be tempted to wonder if our dream provided us with a psychic glimpse into the future despite the fact that the match was close but not exact.

People do not expect paranormal sources of information to be 100 percent accurate. In fact, if they appear to be absolutely spot on, this may arouse suspicion. Nowhere is this better illustrated than in the way that true believers will often try very hard to find a way to interpret a clearly incorrect statement in a psychic reading as somehow “near enough” to be counted as accurate. For example, during a stage performance the medium may say, “I’m getting the name Elizabeth? Does that name mean something to someone in the audience?” In the unlikely event that no one replies positively (and, as we’ve seen, even highly unlikely events do occur), the medium may go on, “Elizabeth? Or Lizzy? Liz? Betty, perhaps?” If an audience member should timidly suggest, “Well, my dead grandmother’s name was Lisa?” both medium and audience will happily accept that as “close enough.”

A Brief Detour: Population Stereotypes

There is a hidden cause behind a fun little demonstration of an ostensibly paranormal experience that I often include in public talks on anomalistic psychology, especially when I have a reasonably large audience. I explain to my audience that an important part of proper skepticism is to always be open to the possibility that you may be wrong. In that spirit, I tell the audience that I would like to do a little experiment with them to see just how psychic they are. I tell them that I am going to try to telepathically send a simple message from my mind to theirs. “I’m thinking of a number between one and ten,” I say. “Not three, because that’s too obvious. I want you to make a mental note of the first number that comes into your mind now!”

I then explain that, with such a large audience, we would expect around 10 percent of them to guess the number correctly just by chance, so we should only get ecstatically excited if considerably more than 10 percent of the audience get it right. I then, apparently somewhat nervously, ask, “Did anybody think of the number seven? If you did, raise your hand.” With a large audience, I can, in fact, be very confident that around a third of them will put up their hand.

Feigning surprise, I will try another, slightly more complicated example. “This time I’m thinking of a two-digit number between one and fifty. They are both odd digits and they are not the same. So, it could be fifteen—one and five, both odd digits, not the same—but it could not be eleven—both odd digits but they are the same. What is the first number that fits that description that comes into your mind now?”

I then ask, as if expecting no one to have got it right this time, “Did anyone think of the number thirty-seven?” Once again, about a third of the audience will put up their hand. I will then add, “Did anyone think of thirty-five?” About a further quarter of the audience will raise their hand. “Sorry, that was my fault,” I explain. “I thought of thirty-five first and then I changed my mind.”

What is going here? Any audience members who believe in ESP may well think that it has just been demonstrated. More skeptical members may be at a loss to explain what they have seen (and possibly directly experienced). Is it possible that I had simply conspired with all those members of the audience who got it right by telling them in advance to raise their hands in response to my questions? That would seem unlikely. Was it just a coincidence that so many more people guessed correctly than would be expected on the basis of chance alone? Again, possible but extremely unlikely.

The actual explanation is a phenomenon that psychologists refer to as population stereotypes.22 For most people faced with this task, when they are asked to make a mental note of the first number that comes into their head, they assume this is pretty much a random process. Therefore, they expect the frequencies of response to be more or less equal across the range of response options. In fact, this is not what happens. Responses tend to cluster in reliable and predictable ways, especially with large audiences.

In the first example, about a third of people will choose seven regardless of whatever number I may be thinking of (especially as I have ruled out three as a valid response, which otherwise would also be a popular choice). In the second example, about a third will pick thirty-seven and about a further quarter will choose thirty-five. Note that in neither example do the response rates approach 100 percent, but that is not a problem as people do not expect telepathy to be 100 percent reliable.

There are several other examples of population stereotypes that could be used to fool (at least some of) the unwary that you possess amazing telepathic powers. Tell them your telepathic target is two simple geometric forms, one inside the other. Around 60 percent will choose circle and triangle. Tell them you are thinking of a simple line drawing. Around 10–12 percent will draw a little house. It makes for a fun demonstration of the fact that not everything that looks like paranormal actually is. But would anyone ever seriously try to pass off such a demonstration as really involving telepathy?

The answer is yes. For example, in the mid-1990s Uri Geller took part in a TV program called Beyond Belief23, presented by David Frost, in which, it was claimed, various paranormal phenomena would be demonstrated live for the millions of viewers at home. I was one of those viewers. Uri demonstrated his alleged telepathic powers by supposedly transmitting a message to viewers. Uri had chosen one of four symbols that were presented at the bottom of the screen in the following order: square, star, circle, cross. As the camera slowly zoomed in on his face, Uri said: “I’m visualizing the symbol in my mind . . . and you people at home, millions of you, I’m actually communicating now with millions of people, maybe eleven, twelve, thirteen million people. Here goes. I’m transmitting it to you. I’m visualizing it. Open up your minds. Try to see it. Try to feel it. I’m really strongly passing it to you! One more time . . . okay.” By this point, the upper half of Uri’s face pretty much filled the screen, with the four symbols still displayed across the bottom of the screen. Viewers were instructed to phone in using one of four different numbers to indicate their guess at Uri’s choice of symbol. Over 70,000 viewers did so.

My days of believing that Uri really did possess amazing psychic abilities were long gone by this stage, and I was therefore watching from the perspective of an informed skeptic. It was pretty easy to come up with nonparanormal explanations for all of the demonstrations featured in the program. With respect to this particular demonstration, I was rather pleased with myself for not only stating in advance what Uri’s telepathic target would be but for also correctly stating the order of popularity of the remaining symbols. It was very lucky for Uri that he chose to “telepathically” transmit the star symbol. It was by far the most popular choice of the viewers, with 47 percent of them indicating that this was the symbol that they had “received.” The second most popular was the circle, with 32 percent of the “votes,” followed by the cross (12 percent) and the square (10 percent). If the guesses were completely random, we would expect about 25 percent for each option, so 47 percent choosing the same symbol as Uri is a massively statistically significant outcome. The probability that almost half of the callers chose this symbol just by chance is astronomically low. So, was this really strong evidence of Uri’s psychic powers?

Readers who are familiar with common techniques used to test for ESP will have recognized that the four symbols used in Uri’s demonstration are taken from the five symbols used on Zener cards (the missing symbol is three wavy lines; figure 8.1). The cards are named after the person who designed them, perceptual psychologist Karl Zener. A full deck consists of twenty-five cards, five of each design. In a test of telepathy, a “sender” would take each card from a shuffled deck in turn and attempt to telepathically transmit the image on the card to a “receiver.” The receiver would record their guess of which card the sender was looking at. By chance alone, we would expect around five of the receiver’s guesses to be correct. If the receiver scores significantly more than five, this might be taken as evidence of ESP. However, it has been known for over eight decades that people are more likely to guess certain symbols compared to others. Back in 1939, Frederick Lund asked 596 people to each generate a random sequence of five symbols from the Zener set.24 By far the most popular symbol was—you’ve guessed it—the star, accounting for 32 percent of the responses compared to the 20 percent that would be expected by chance alone. So, as I said, it really was lucky for Uri that he chose the star as his telepathic target (assuming that it was just luck).

Figure 8.1

The five symbols used on Zener cards.

Probabilistic Reasoning and Paranormal Belief

As stated, we are all very poor at dealing with probabilities in everyday life. This is one aspect of what psychologists call probabilistic reasoning, but this term covers more than just our (lack of) understanding of probabilities per se. Other aspects of probabilistic reasoning that typically cause us problems include the base rate fallacy as well as our inadequate understanding of the true nature of randomness.25

The base rate fallacy (also known as base rate neglect) can be demonstrated in many different contexts. In the words of Maya Bar-Hillel, “The base-rate fallacy is people’s tendency to ignore base rates in favor of, e.g., individuating information (when such is available), rather than integrate the two. This tendency has important implications for understanding judgment phenomena in many clinical, legal, and social-psychological settings.”26

A classic demonstration of this tendency is the cabs problem:

. . . where people are told that 85% of the cabs in a city are blue and 15% are green. They also are told that there was a hit and run accident at night and that a witness, who has been shown to be able to correctly identify the color of the cab at night 80% of the time, identified the cab as being green. People are then asked how likely it is that the cab is green, and many report an 80% likelihood. However, this conclusion does not consider the base rate of the two colours of cabs within the city. . . . Specifically, it is important to consider both the individual descriptive information about the witness’s accuracy and the base rate of green cabs. Based on both, the correct probability of the cab being green is 41% because, even though the witness is 80% accurate and identified the cab as green, the witness is still wrong 20% of the time; thus, due to the low base rate of green cabs (15%), it is more likely that the cab was blue than green.27

Our generally poor appreciation of the true nature of randomness is vividly demonstrated by the fact that, when asked to do so, people are incapable of generating true random sequences without the aid of external physical devices such as dice.28 When trying to generate truly random sequences, people show a strong tendency to avoid consecutive repetition of the same element. In fact, of course, each element in a random sequence is entirely independent of all other elements in the sequence, including those that come immediately before or after it. Thus, in a random sequence of digits from zero to nine, a six is just as likely to be followed by a six as any other digit. Runs of the same digit are to be expected.

The same repetition-avoidance bias occurs when we try to judge whether some external process, such as a roulette wheel, is truly random or not. One notorious example of this is known as the gambler’s fallacy. This is the tendency that some people have to assume that a black number, as opposed to a red number, is more likely to occur on the wheel after a long run of red numbers. In fact, of course, black and red are still equally likely.

Given our problems with everyday probabilistic reasoning, the question naturally arises of whether believers in the paranormal may be somewhat worse than nonbelievers in this respect, thus making them more likely to seek a paranormal explanation for various experiences where in fact no such explanation is required. Several studies have addressed this possibility, although a comprehensive review of this literature is beyond the scope of the current chapter.29

A very wide range of tasks are used in these studies, along with different measures of paranormal belief, conditions of testing, and types of participant. Many studies have indeed obtained results supporting the idea that believers in the paranormal show stronger deficits in probabilistic reasoning compared to nonbelievers.30 Others have found such effects for some probabilistic reasoning tasks but not others,31 and at least one failed to find any such effects.32 Taken as a whole, it appears that, in general, believers in the paranormal are somewhat poorer at certain aspects of probabilistic reasoning than nonbelievers, and this is probably one factor that contributes to the believers’ tendency to interpret more experiences, such as coincidences, as involving paranormal forces. However, it may well be that misperceptions of chance per se are less important than other aspects of probabilistic reasoning, such as poor appreciation of the true nature of randomness and an increased propensity to find connections between separate events.

The latter has been said to be a factor in explaining another type of reasoning error often associated with paranormal belief: the conjunction fallacy, first described by Amos Tversky and Daniel Kahneman.33 Consider the following:

Linda is 31 years old, single, 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.

Which is more probable?

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

You may find that you, in line with 85 percent to 90 percent of undergraduate students, feel that the second option is more probable. But this is logically impossible. If you think about it for a second, you will realize that the set of bank tellers who are also active feminists is a subset of the set of all bank tellers; therefore, the second option cannot be more probable. Tversky and Kahneman argued that people are misled because they rely on the representativeness heuristic in making their decision. Because the description of Linda corresponds to a stereotypical description of a feminist, this has an undue influence on our thinking, causing us to commit a logical error.

The idea that believers in the paranormal might have an increased propensity to find connections between separate events lends itself naturally to the possibility that they may be more susceptible to the conjunction fallacy insofar as they may tend to overestimate the probability of conjunctive (co-occurring) events compared to component (singular) events. This hypothesis has been supported across a number of studies, particularly those carried out by Paul Rogers and colleagues.34 It should be noted that the strength of the effect varies with such factors as the content of the hypothetical events (e.g., paranormal versus nonparanormal, confirmatory versus nonconfirmatory) and the measure used to measure paranormal belief. However, Neil Dagnall and colleagues argue that overall misperception of the nature of randomness is more strongly associated with paranormal belief than is susceptibility to the conjunction fallacy.35

Before we leave the topic of coincidences, it is important to point out that our tendency to pay particular attention to the co-occurrence of matching events has been hugely important to our evolutionary history. Although we may sometimes be tempted to infer a causal connection between two co-occurring events when no such connection exists, it is clearly the case that sometimes there really is a causal connection.

This important point has been emphasized by Mark Johansen and Magda Osman.36 It would be a big mistake to dismiss all reports of apparent coincidences as being due to nothing more than chance. Johansen and Osman define coincidences as “surprising pattern repetitions that are observed to be unlikely by chance but are nonetheless ascribed to chance since the search for causal mechanisms has not produced anything more plausible than mere chance.”37

They argue that the experience of coincidences should not be viewed as an example of human irrationality but instead as a fundamental consequence of rational cognition. We can only describe the relationship between two events as truly coincidental if we have first excluded potential causal links. Indeed, sometimes scientific breakthroughs, such as the discovery of penicillin by Sir Alexander Fleming in 1928, are based on a serendipitous observation that, fortunately for humanity, was not dismissed as “just a coincidence.”

As will be discussed further in the next chapter, many of the cognitive biases that are associated with paranormal belief have evolved because, on balance, they conferred more advantages than disadvantages upon our species in increasing our chances of surviving long enough to pass on our genes to the next generation. We may sometimes be misled into thinking that there is a causal relationship between two events when in fact there isn’t—but this is outweighed by the advantages of being able to detect such relationships when they really are there to be detected.