Low odds, but there are 10 million people
watching
the show.
Richard-Dawkins-Unweaving-the-Rainbow
Moreover, the other and apparently more scientific way to gain useful knowledge, learning by trial and error, is often a bad idea because the errors are too costly.
If your mother tells you never to paddle in the lake because of the crocodiles, it is no good coming over all sceptical and scientific and 'adult' and saying, 'Thank you mother, but I prefer to put it to the experimental test.
' Too often, such experiments would be terminal.
It is easy to see why natural selection - the survival of the fittest - might penalize an experimental and sceptical turn of mind and favour simple credulity in children.
But this has an unfortunate by-product which can't be helped. If your parents tell you something that is not true, you must believe that, too. How could you not? Children are not equipped to know the difference between a true warning about genuine dangers and a false warning about going blind, say, or going to hell, if you 'sin'. If they were so equipped, they wouldn't need warnings at all. Credulity, as a survival device, comes as a package. You believe what you are told, the false with the true. Parents and elders know so much, it is natural to assume that they know everything and natural to believe them. So when they tell you about Father Christmas coming down the chimney, and about faith 'moving mountains', of course you believe that, too.
Children are gullible because they need to be if they are to fulfil their 'caterpillar' role in life. Butterflies have wings because their role is to locate members of the opposite sex and spread their offspring to new food plants. They have modest appetites satisfied by occasional sips of nectar. They eat little protein by comparison with caterpillars, which constitute the growing stage in the life history. Juvenile animals in general have the role of preparing to become successfully reproducing adults. Caterpillars are there to feed as rapidly as possible in order to chrysalize into flying, reproducing, dispersing adults. To this end they have no wings but instead have stout munching jaws and voracious, single-minded appetites.
Human children need to be credulous for a similar reason. They are information caterpillars. They are there to become reproducing adults, in a sophisticated, knowledge-based society. And by far the most important source of their information diet is their elders, above all their parents. For the same kind of reason as caterpillars have chumbling, hoovering jaws for sucking up cabbage flesh, human children have wide open ears and eyes, and gaping, trusting minds for sucking up language and other knowledge. They are suckers for adult knowledge. Tidal waves of data, gigabytes of wisdom flood through the portals of the infant skull, and most of it originates in the culture built up by parents and generations of ancestors. It is important, incidentally, not to take the caterpillar analogy too far. Children change gradually into adults, not suddenly, as caterpillars metamorphose into butterflies.
I remember once trying gently to amuse a six-year old child at Christmas time by reckoning up with her how long it would take Father Christmas to go down all the chimneys in the world. If the average chimney is 20 feet long and there are, say, 100 million houses with children, how fast, I wondered aloud, would he have to whizz down each chimney in order to finish the job by dawn on Christmas Day? He'd hardly have time to tiptoe noiselessly into each child's bedroom, would he, since he'd necessarily be breaking the sound barrier? She saw the point and realized that there was a problem, but it didn't worry her in the least. She dropped the subject without pursuing it. The obvious possibility that her parents had been telling falsehoods never seemed to cross her mind. She wouldn't have put it in these words but the implication was that, if the laws of physics rendered Father Christmas's feat impossible, so much the worse for the laws of physics. It was enough that her parents had told her he went down all the chimneys during the few hours of Christmas Eve. It must be so because Mummy and Daddy said it was.
My contention is that trusting credulity may be normal and healthy in a child but it can become an unhealthy and reprehensible gullibility in an adult. Growing up, in the fullest sense of the word, should include the cultivation of a healthy scepticism. An active readiness to be deceived can be called childish because it is common - and defensible - among children. I suspect that its persistence in adults stems from a hankering after, indeed a pining for, the lost securities and comforts of childhood. The point was well put in 1986 by that great writer of popular science and science fiction Isaac Asimov: 'Inspect every piece of pseudoscience and you will find a security blanket, a thumb to suck, a skirt to hold. ' Childhood is, for many people, a lost Arcadia, a kind of heaven, with its certainties and its securities, its fantasies of flying to the Never Never Land, its bedtime stories before we drifted off to the Land of Nod in the arms of Teddy Bear. With hindsight, the years of childish innocence may pass too soon. I love my parents for taking me for a ride, high as a kite,
through the treetops; and for telling me about the Tooth Fairy and Father Christmas, about Merlin and his spells, about baby Jesus and the Three Wise Men, All these stories enrich childhood and, together with so many other things, help to make it, in memory, a time of enchantment.
The adult world may seem a cold and empty place, with no fairies and no Father Christmas, no Toyland or Narnia, no Happy Hunting Ground where mourned pets go, and no angels - guardian or garden variety. But there are also no devils, no hellfire, no wicked witches, no ghosts, no haunted houses, no daemonic possession, no bogeymen or ogres. Yes, Teddy and Dolly turn out not to be really alive. But there are warm, live, speaking, thinking, adult bedfellows to hold, and many of us find it a more rewarding kind of love than the childish affection for stuffed toys, however soft and cuddly they may be.
Not to grow up properly is to retain our 'caterpillar' quality from childhood (where it is a virtue) into adulthood (where it becomes a vice). In childhood our credulity serves us well. It helps us to pack, with extraordinary rapidity, our skulls full of the wisdom of our parents and our ancestors. But if we don't grow out of it in the fullness of time, our caterpillar nature makes us a sitting target for astrologers, mediums, gurus, evangelists and quacks. The genius of the human child, mental caterpillar extraordinary, is for soaking up information and ideas, not for criticizing them. If critical faculties later grow it will be in spite of, not because of, the inclinations of childhood. The blotting paper of the child's brain is the unpromising seedbed, the base upon which later the sceptical attitude, like a struggling mustard plant, may possibly grow. We need to replace the automatic credulity of childhood with the constructive scepticism of adult science.
But I suspect an additional problem. Our story of the child as
information caterpillar was too simple. The programming of the child's credulity has a twist which, until we understand it, is almost paradoxical. Let us go back to our picture of the child needing to absorb information from the previous generation as swiftly as possible. What if two adults, say your mother and your father, give you contradictory advice? What if your mother tells you that all snakes are deadly and you must never go near them, but next day your father tells you that all snakes are deadly except green ones and you can keep a green snake as a pet? Both pieces of advice may be good. The mother's more general advice has the desired effect of protecting you against snakes, even though it is too sweeping when it comes to green snakes. The father's more discriminating advice has the same protective effect and is in some ways better, But it could be fatal if carried, unrevised, to a distant country. In any case, to the young child the contradiction between the two might be dangerously confusing. Parents often make strenuous efforts not to contradict one another, and
they are probably wise to do so. But natural selection, in 'designing' credulity, would need to build in a way of coping with contradictory advice. Perhaps a simple override rule, such as 'Believe whichever story you heard first. ' Or 'Believe mother rather than father, and father rather than other adults in the population. '
Sometimes the advice from parents is specifically aimed against credulity towards other adults in the population. The following is a piece of advice that parents need to give their children: 'If any adult asks you to come with him and says that he is a friend of your parents, don't believe him, however nice he seems and even (or especially) if he offers you sweets. Only go with an adult that you and your parents already know, or who is wearing a policeman's uniform. ' (A charming story recently appeared in the English newspapers in which Queen Elizabeth the Queen Mother, aged 97, told her chauffeur to stop the car when she noticed a crying child who was apparently lost. The kind old lady got out to comfort the little girl and offered to take her home. 'I can't,' wailed the child, 'I'm not allowed to talk to strangers. ') A child is called upon to exercise the exact opposite of credulity in some circumstances: a tenacious persistence in believing an earlier adult statement in the face of what may be a tempting and plausible - but contradictory - later statement.
On their own, then, the words 'gullible' and 'credulous' are not quite right for children. Truly credulous people believe whatever they have most recently been told, even if this contradicts what others have told them before. The quality of childhood that I am trying to pin down is not pure gullibility but a complex combination of gullibility coupled with its opposite - stubborn persistence in a belief, once acquired. The full recipe, then, is extreme early gullibility followed by equally obstinate subsequent unshakeability. You can see what a devastating combination this could be. Those old Jesuits knew what they were about: 'Give me the child for his first seven years, and I'll give you the man. '
7
UNWEAVING THE UNCANNY
. . . though no great ministering reason sorts Out the dark mysteries of human souls
To clear conceiving. . .
JOHN KEATS, 'Sleep and Poetry' (1817)
The eminent fertility specialist Robert Winston imagines the following advertisement, placed in the newspaper by an unscrupulous quack
doctor, aimed at people who want their next baby to be, say, a son (the sexism underlying this assumption is not mine but could be found unquestioned all over the ancient world, and still in many places today). 'Send ? 500 for my patent recipe to make your baby a boy. Money refunded in full if I fail. ' The money back guarantee is intended to establish confidence in the method. In fact, of course, since boys turn up anyway on approximately 50 per cent of occasions, the scheme would be a nice little earner. Indeed, the quack could safely offer compensation of, say, ? 250 for every girl born, over and above the money back guarantee. He would still show a tidy profit in the long run.
I used a similar illustration in one of my Royal Institution Christmas Lectures in 1991. I said I had reason to believe that among my audience was a psychic, clairvoyant individual, capable of influencing events purely by the power of thought. I would try to flush this individual out. 'Let's first establish,' I said, 'whether the psychic is in the left half or the right half of the lecture hall. ' I invited everybody to stand up while my assistant tossed a coin.
Everybody on the left of the hall was asked to 'will' the coin to come down heads. Everybody on the right had to will it to be tails. Obviously one side had to lose, and they were asked to sit down. Then those that remained were divided into two, with half 'willing' heads and the other half tails. Again the losers sat down. And so on by successive halvings until, inevitably, after seven or eight tosses, one individual was left standing. 'A big round of applause for our psychic' He must be psychic, mustn't he, because he successfully influenced the coin eight times in a row?
If the lectures had been televised live, instead of recorded and broadcast later, the demonstration would have been much more impressive. I'd have asked everybody who watched it whose surname begins before J in the alphabet to 'will' heads and the rest tails. Whichever half turned out to contain the 'psychic' would have been divided in half again, and so on. I'd have asked everybody to keep a written record of the order of their 'willings'. With two million viewers, it would have taken about 21 steps to narrow down to a single individual. To be on the safe side I'd have stopped a bit short of 21 steps. At, say, the eighteenth step I'd have invited anybody still in the game to phone in. There would have been quite a few and, with luck, one would phone. This individual would then have been invited to read out his/her written record: which would have matched the official record. So this one individual succeeded in influencing 18 successive tosses of a coin. Gasps of admiration. But admiration for what? Nothing but pure luck. I don't know if that experiment has been done. Actually, the trick here is so obvious it probably wouldn't fool many people. But how about the following?
A well-known 'psychic' goes on television, a lucrative engagement fixed up over lunch by his publicity agent. Staring out of ten million screens with hypnotically smouldering eyes (nice job by Make-up and Lighting), our imaginary seer intones that he feels a strange, spiritual rapport, a vibrating resonance of cosmic energy, with certain members of his audience. They will be able to tell who they are because, even as he utters his mystic incantation, their watches will stop. After only a brief pause, a telephone on his table rings and an amplified voice in awed tones announces that its owner's watch stopped dead within seconds of the clairvoyant's words. The caller adds that she had a premonition that this was going to happen even before she looked down at her watch, for something in her hero's burning eyes seemed to speak directly to her soul. She felt the 'vibrations' of 'energy'. Even as she is speaking, a second telephone rings. Yet another watch has stopped.
A third caller's grandfather clock stopped - surely a weightier feat than stopping a little watch whose delicate hairspring would naturally be more susceptible to psychic forces than the massive pendulum of the grandfather! Another viewer's watch actually stopped a little before the celebrated mystic made his pronouncement - is this not an even more impressive feat of psychic control? Yet another watch has been more impatiently susceptible to occult forces. It had stopped a whole day before, at the very moment when its owner looked at the famous mystic's photograph in the newspaper. The studio audience gasps its appreciation. This, surely, is psychic power beyond all scepticism, for it happened a whole day early! 'There are more things in heaven and earth, Horatio . . . '
What we need is less gasping and more thinking. This chapter is about how to take the sting out of coincidence by quietly sitting down and calculating the likelihood that it would have happened anyway. In the course of this, we shall discover that to disarm apparently uncanny coincidences is more interesting than gasping over them anyway.
Sometimes the calculation is easy. In a previous book I gave away the number of the combination lock on my bicycle. I felt safe in doing so because obviously my books would never be read by the kind of person who would steal a bicycle. Unfortunately somebody did steal it, and I now have a new lock with a new number, 4167. I find this number easy to remember. 41 is imprinted in my memory as the arbitrary code used to identify my clothes and shoes at boarding school. 67 is the age at which I am due to retire. Obviously there is no interesting coincidence here: whatever the number had been, I'd have searched my life for a mnemonic recipe and I'd have found it. But mark the sequel. On the day of writing this, I received from my Oxford college a letter saying:
Each person authorized to use the photocopiers is issued with a personal code number which permits access. Your new number is 4167.
My first thought was that I'd undoubtedly lose this piece of paper (I quickly lost its equivalent last year) and I must immediately think of a formula to fix it in my memory. Something similar to the mnemonic by which I remember my bicycle combination, perhaps? So I looked again at the number on the letter and, to borrow a neat line from Fred Hoyle's science fiction novel The Black Cloud, the figures on the piece of paper seemed to swell to a gigantic size. I didn't need a new mnemonic. The number was identical. I rushed to tell my wife of the amazing coincidence, but on more sober reflection I shouldn't have bothered.
The odds of this happening by chance alone are easily calculated. The first digit could have been anything from 0 to 9. So there is a one in 10 chance of getting a 4 and matching the bicycle lock. For each of these ten possibilities, the second digit could have been anything from 0 to 9, so again there is a one in 10 chance of matching the bike lock's second dial. The odds of matching the first two digits is therefore one in 100 and, following the logic through the other two digits, the odds of matching all four digits of the bicycle lock is one in 10,000. It is this large number that is our protection against theft.
The coincidence is impressive. But what should we conclude? Has something mysterious and providential been going on? Have guardian angels been at work behind the scenes? Have lucky stars swum into Uranus? No. There is no reason to suspect anything more than simple accident. The number of people in the world is so large compared with 10,000 that somebody, at this very moment, is bound to be experiencing a coincidence at least as startling as mine. It just happens that today was my day to notice such a coincidence. It isn't even an added coincidence that it happened to me on this particular day, while I was writing this chapter. I had in fact written the first draft of the chapter some weeks ago. I reopened it today, after the coincidence occurred, in order to insert this anecdote. I shall surely reopen it many times to revise and polish, and I shall not remove the references to 'today': they were accurate when written. This is another way in which we habitually inflate the impressiveness of coincidence in order to make a good story.
We can do a similar calculation for the television guru whose psychic miasma seemed to stop people's watches, but we'll have to use estimates rather than exact figures. Any given watch has a certain low probability of stopping at any moment. I don't know what this probability is, but here's the kind of way in which we could come to an estimate. If we take just digital watches, their battery typically runs out within a year. Approximately, then, a digital watch stops once per year. Presumably clockwork watches stop more often because people forget to wind them
and presumably digital watches stop less often because people sometimes remember to renew the battery ahead of time. But both kinds of watches probably stop as often again because they develop faults of one kind or another. So, let our estimate be that any given watch is likely to stop about once a year. It doesn't matter too much how accurate our estimate is. The principle will remain.
If somebody's watch stopped three weeks after the spell was cast, even the most credulous would prefer to put it down to chance. We need to decide how large a delay would have been judged by the audience as sufficiently simultaneous with the psychic's announcement to impress. About five minutes is certainly safe, especially since he can keep talking to each caller for a few minutes before the next call ceases to seem roughly simultaneous. There are about 100,000 five-minute periods in a year. The probability that any given watch, say mine, will stop in a designated five-minute period is about 1 in 100,000.
Low odds, but there are 10 million people watching the show. If only half of them are wearing watches, we could expect about 25 of those watches to stop in any given minute. If only a quarter of these ring in to the studio, that is 6 calls, more than enough to dumbfound a naive audience. Especially when you add in the calls from people whose watches stopped the day before, people whose watches didn't stop but whose grandfather clocks did, people who died of heart attacks and their bereaved relatives phoned in to say that their 'ticker' gave out, and so on. This kind of coincidence is celebrated in the delightfully sentimental old song, 'Grandfather's Clock:'
Ninety years without slumbering. Tick, lock, tick, tock.
His life seconds numbering. Tick, tock, tick, tock.
It stopped . . . short. . . never to go again When the old man died.
Richard Feynman, in a 1963 lecture published posthumously in 1998, tells the story of how his first wife died at 9. 22 in the evening and the clock in her room was later found to have stopped at exactly 9. 22. There are those who would revel in the apparent mystery of this coincidence and feel that Feynman has taken away something precious when he gives us a simple, rational explanation of the mystery. The clock was old and erratic and was in the habit of stopping if tilted out of the horizontal. Feynman himself frequently repaired it. When Mrs Feynman died, the
nurse's duty was to record the exact time of death. She moved over to the clock, but it was in dark shadow. In order to see it, she picked it up - and tilted its face towards the light . . . The clock stopped. Is Feynman really spoiling something beautiful when he tells us what is surely the true - and very simple - explanation? Not for my money. For me, he is affirming the elegance and beauty of an orderly universe in which clocks stop for reasons, not to titillate human sentimental fancy.
At this point, I want to invent a technical term, and I hope you'll forgive an acronym. PETWHAC stands for Population of Events That Would Have Appeared Coincidental. Population may seem an odd word, but it is the correct statistical term. I won't keep using capital letters because they stand so unattractively on the page. Somebody's watch stopping within ten seconds of the psychic's incantation obviously belongs within the petwhac, but so do many other events. Strictly speaking, the grandfather clock's stopping should not be included. The mystic did not claim that he could stop grandfather clocks. Yet when somebody's grandfather clock did stop, they immediately telephoned in because they were, if anything, even more impressed than they would have been if their watch had stopped. The odd misconception is fostered that the psychic is even more powerful since he didn't even bother to mention that he could stop grandfather clocks, too! Similarly, he said nothing about watches stopping the day before or grandfathers' tickers suffering cardiac arrests.
People feel that such unanticipated events belong in the petwhac. It looks to them as though occult forces must have been at work. But when you start to think like this, the petwhac becomes really quite large, and therein lies the catch. If your watch stopped exactly 24 hours earlier, you would not have to be unduly gullible to embrace this event within the petwhac. If somebody's watch stopped exactly seven minutes before the spell, this might impress some people because seven is an ancient mystic number. And the same presumably goes for seven hours, seven days . . . The larger the petwhac, the less we ought to be impressed by the coincidence when it comes. One of the devices of an effective trickster is to make people think exactly the opposite.
By the way, I deliberately chose a more impressive trick for my imaginary psychic than is actually done with watches on television. The more familiar feat is to start watches that have stopped. The television
audience is invited to get up and fetch, out of drawers or attics, watches that have broken down, and hold them while the psychic performs some incantation or does some hypnotic eye work. What is really going on is that the warmth of the hand melts oil that has coagulated and this starts the watch ticking, if only briefly. Even if this happens in only a small proportion of cases, this proportion, multiplied by the large audience, will generate a satisfactory number of dumbfounded telephone calls. Actually,
as Nicholas Humphrey explains in his admirable expose of supernaturalism Soul Searching (1995), it has been demonstrated that more than 50 per cent of broken watches start, at least momentarily, if they are held in the hand.
Here's another example of a coincidence, where it is clear how to calculate the odds. We shall use it to go on and see how odds are sensitive to changing the petwhac. I once had a girlfriend who had the same birth date (though not in the same year) as my previous girlfriend. She told a friend of hers who believed in astrology, and the friend triumphantly asked how I could possibly justify my scepticism in the face of such overwhelming evidence that I had unwittingly been brought together with two successive women on the basis of their 'stars'. Once again, let's just think it through quietly. It is easy to calculate the odds that two people, chosen entirely at random, will have the same birthday. There are 565 days in the year. Whatever the birthday of the first person, the chance that the second will have the same birthday is 1 in 365 (forgetting leap years). If we pair people off in any particular way, such as taking the successive women friends of any one man, the odds that they will share their birthday are 1 in 365. If we take ten million men (less than the population of Tokyo or Mexico City), this apparently uncanny coincidence will have happened to more than 27,000 of them!
Now let's think about the petwhac and see how the apparent coincidence becomes less impressive as it swells. There are many other ways in which we could pair people off and still end up noticing an apparent coincidence. Two successive girlfriends with the same surname, although unrelated, for instance. Two business partners with the same birthday would also come within the petwhac; or two people with the same birthday sitting next to one another on an aeroplane. Yet, in a well- loaded Boeing 747, the odds are actually better than 50 per cent that at least one pair of neighbours will share a birthday. We don't usually notice this because we don't look over each other's shoulders as we fill in those tedious immigration forms. But if we did, somebody on most flights would go away muttering darkly about occult forces.
The birthday coincidence is famously phrased in a more dramatic way. If you have a roomful of only 25 people, mathematicians can prove that the odds are just greater than 50 per cent that at least two of them will share the same birthday. Two readers of an earlier draft of the book asked me to justify this astonishing statement. It's easier to calculate the odds that there won't be any shared birthdays and subtract from one. Forget about leap years because they're more trouble than they're worth. Suppose I bet you that with 25 people in a room, at least two will share a birthday. You bet, for the sake of argument, that there'll be no shared birthdays. We're going to do the calculation by working up to 23 people gradually,
starting with just one person in the room, and adding people one at a time. If at any point a match is found, I've won the bet, we stop the game and don't bother to add any more people. If we get to 23 people and there's still no match so far, you win the bet.
When the room contains only the first person, whom we may as well call A, the chance of 'no match so far' is trivially 1 (365 out of 365 chances). Now add a second person, B. The chance of a match is now one in 365. So the odds of 'no match so far' when B has joined A in the room are 364/365. Now add a third person, C, There's a one in 365 chance that C matches A and a one in 365 chance that C matches B, therefore the chance that he matches neither A nor B is 363/365 (he can't match both, because we already know that A doesn't match B). To get the total odds
of 'no match so far' we have to take this 363/365 and multiply it by the odds against a match in the previous round (s), in this case by 364/365. The same reasoning applies when we add the fourth person, D. The total odds of 'no match so far' are now 364/365 X 363/365 X 362/365. And so on until all 23 people are in the room. Each new person adds a new term that we have to bring in to the running multiplication sum, to compute 'no match so far'.
If you multiply this out for 23 terms (you have to go on down to 343/365) the answer comes to about 0. 49. This is the chance that there will not be any shared birthdays in the room. So there's a slightly greater than even chance that at least one pair of individuals in a committee of 23 will share a birthday. Most people's intuition would encourage them to bet against such a coincidence. But they'd be wrong. It is this kind of intuitive error that in general bedevils our assessment of 'uncanny' coincidences.
Here's an actual coincidence where, although it is a little harder, we can make a stab at estimating the odds approximately. My wife once bought for her mother a beautiful antique watch with a pink face. When she got it home and peeled off the price label she was amazed to find, engraved on the back of the watch, her mother's own initials, M. A. B. Uncanny? Eerie? Spine-crawling? Arthur Koestler, the famous novelist, would have read much into it. So would C. G. Jung, the widely admired psychologist and inventor of the 'collective unconscious', who also believed that a bookcase or a knife might be induced by psychic forces to explode spontaneously with a loud report. My wife, who has more sense, merely thought the coincidence of initials remarkably convenient and sufficiently amusing to justify telling the story to me - and here I am now telling it to a wider audience.
So, what really are the odds against a coincidence of this magnitude? We can begin by calculating them in a naive way. There are 26 letters in the
alphabet. If your mother has three initials and you find a watch engraved with three letters at random, the odds that the two will coincide is 1/26 x 1/26 x 1/26, or one in 17,576. There are about 55 million people in Britain. If every one of them bought an antique engraved watch we'd expect more than 3,000 of them to gasp with amazement when they discovered that the watch already bore their mother's initials.
But the odds are actually better than this. Our naive calculation made the incorrect assumption that each letter has a probability of 1/26 of being somebody's initial. This is the average probability for the alphabet as a whole, but some letters, such as X and Z, have a smaller probability. Others, including M, A and B, are commoner: think how much more impressed we'd be if the coinciding initials had been X. Q. Z. We can improve our estimate of odds by sampling a telephone directory. Sampling is a respectable way of estimating something that we cannot count directly. The London directory is a good place to sample because it is large and London happens to be where my wife bought the watch and where her mother lived. The London telephone directory contains about 85,060 column inches, or about 1. 34 column miles, of private citizens' names. Of these, about 8,110 column inches are devoted to the Bs. This means that about 9. 5 per cent of Londoners have a surname beginning with B - much more frequent than the figure for an average letter: 1/26, or 3. 8 per cent.
So, the probability that a randomly chosen Londoner would have a surname beginning with B is about 0. 095 (~ 9. 5 per cent). What about
the corresponding probabilities that the forenames will begin with M or A? It would take too long to count forename initials right through the telephone book, and there'd be no point since the telephone book is itself only a sample. The easiest thing to do is take a subsample where forename initials are conveniently arranged in alphabetical order. This is true of the listings within any one surname. I shall take the commonest surname in England - Smith - and look at what proportion of the Smiths are M. Smith and what proportion are A. Smith. It is a reasonable hope that this will be approximately representative of the probabilities of forename initials for Londoners generally. It turns out that there are rather more than 20 column yards of Smiths altogether. Of these, 0. 075 them (53. 6 column inches) are M. Smiths. The A. Smiths fill 75-4 column inches, representing 0. 102 of all the Smiths,
If you are a Londoner and you have three initials, therefore, the chances of your initials being M. A. B. in that order are approximately 0. 102 X 0. 073 0-095 or about 0. 0007. Since the population of Britain is 55 million, this should mean that about 38,000 of them have the initials M. A. B. , but only if everybody among those 55 million has three initials. Obviously not everybody does but, looking down the telephone directory
again, it seems that at least a majority do. If we make the conservative assumption that only half of British people have three initials, that still means that more than 19,000 British people have identical initials to my wife's mother. Any one of them could have bought that watch and gasped with astonishment at the coincidence. Our calculation has shown that there is no reason to gasp.
Indeed, when we think harder about the petwhac, we find that we have even less right to be impressed. M. A. B. were the initial letters of my wife's mother's maiden name. Her married initials of M. A. W. would have seemed just as impressive had they been found on the watch. Surnames beginning with W are nearly as common in the telephone book as those beginning with B. This consideration approximately doubles the petwhac, by doubling the number of people in the country who would have been deemed, by a coincidence hunter, to have 'the same initials' as my wife's mother.
Moreover, if somebody bought a watch and found it to be engraved not with her mother's initials but with her own, she might consider it an even greater coincidence and more worthy to be embraced within the (ever-growing) petwhac.
The late Arthur Koestler, as I have already mentioned, was a great enthusiast of coincidences. Among the stories that he recounts in The Roots of Coincidence (1972) are several originally collected by his hero, the Austrian biologist Paul Kammerer (famous for publishing a faked experiment purportedly demonstrating the 'inheritance of acquired characteristics' in the midwife toad). Here is a typical Kammerer story quoted by Koestler:
On September 18, 1916, my wife, while waiting for her turn in the consulting rooms of Prof. Dr J. v. H. , reads the magazine Die Kunst; she is impressed by some reproductions of pictures by a painter named Schwalbach, and makes a mental note to remember his name because she would like to see the originals. At that moment the door opens and the receptionist calls out to the patients: 'Is Frau Schwalbach here? She is wanted on the telephone. '
It probably isn't worth trying to estimate the odds against this coincidence, but we can at least write down some of the things that we'd need to know. 'At that moment the door opens' is a little vague. Did the door open one second after she made the mental note to look up Schwalbach's paintings or 20 minutes? How long could the interval have been, leaving her still impressed by the coincidence? The frequency of the name Schwalbach is obviously relevant: we'd be less impressed if it had been Schmidt or Strauss; more impressed if it had been Twistleton- Wykeham-Fiennes or Knatchbull-Huguesson. My local library doesn't
have the Vienna telephone book, but a quick look in another large Germanic telephone directory, the Berlin one, yields half a dozen Schwalbachs: the name is not particularly common, therefore, and it is understandable that the lady was impressed. But we need to think further about the size of the petwhac. Similar coincidences could have happened to people in other doctors' waiting rooms; and in dentists' waiting rooms, government offices and so on; and not just in Vienna but anywhere else. The quantity to keep bearing in mind is the number of opportunities for coincidence that would have been thought, if they had occurred, just as remarkable as the one that actually did occur.
Now let's take another kind of coincidence, where it is even harder to know how to start calculating odds. Consider the often-quoted experience of dreaming of an old acquaintance for the first time in years and then getting a letter from him, out of the blue, the next day. Or of learning that he died in the night. Or of learning that he didn't die in the night but his father did.
But this has an unfortunate by-product which can't be helped. If your parents tell you something that is not true, you must believe that, too. How could you not? Children are not equipped to know the difference between a true warning about genuine dangers and a false warning about going blind, say, or going to hell, if you 'sin'. If they were so equipped, they wouldn't need warnings at all. Credulity, as a survival device, comes as a package. You believe what you are told, the false with the true. Parents and elders know so much, it is natural to assume that they know everything and natural to believe them. So when they tell you about Father Christmas coming down the chimney, and about faith 'moving mountains', of course you believe that, too.
Children are gullible because they need to be if they are to fulfil their 'caterpillar' role in life. Butterflies have wings because their role is to locate members of the opposite sex and spread their offspring to new food plants. They have modest appetites satisfied by occasional sips of nectar. They eat little protein by comparison with caterpillars, which constitute the growing stage in the life history. Juvenile animals in general have the role of preparing to become successfully reproducing adults. Caterpillars are there to feed as rapidly as possible in order to chrysalize into flying, reproducing, dispersing adults. To this end they have no wings but instead have stout munching jaws and voracious, single-minded appetites.
Human children need to be credulous for a similar reason. They are information caterpillars. They are there to become reproducing adults, in a sophisticated, knowledge-based society. And by far the most important source of their information diet is their elders, above all their parents. For the same kind of reason as caterpillars have chumbling, hoovering jaws for sucking up cabbage flesh, human children have wide open ears and eyes, and gaping, trusting minds for sucking up language and other knowledge. They are suckers for adult knowledge. Tidal waves of data, gigabytes of wisdom flood through the portals of the infant skull, and most of it originates in the culture built up by parents and generations of ancestors. It is important, incidentally, not to take the caterpillar analogy too far. Children change gradually into adults, not suddenly, as caterpillars metamorphose into butterflies.
I remember once trying gently to amuse a six-year old child at Christmas time by reckoning up with her how long it would take Father Christmas to go down all the chimneys in the world. If the average chimney is 20 feet long and there are, say, 100 million houses with children, how fast, I wondered aloud, would he have to whizz down each chimney in order to finish the job by dawn on Christmas Day? He'd hardly have time to tiptoe noiselessly into each child's bedroom, would he, since he'd necessarily be breaking the sound barrier? She saw the point and realized that there was a problem, but it didn't worry her in the least. She dropped the subject without pursuing it. The obvious possibility that her parents had been telling falsehoods never seemed to cross her mind. She wouldn't have put it in these words but the implication was that, if the laws of physics rendered Father Christmas's feat impossible, so much the worse for the laws of physics. It was enough that her parents had told her he went down all the chimneys during the few hours of Christmas Eve. It must be so because Mummy and Daddy said it was.
My contention is that trusting credulity may be normal and healthy in a child but it can become an unhealthy and reprehensible gullibility in an adult. Growing up, in the fullest sense of the word, should include the cultivation of a healthy scepticism. An active readiness to be deceived can be called childish because it is common - and defensible - among children. I suspect that its persistence in adults stems from a hankering after, indeed a pining for, the lost securities and comforts of childhood. The point was well put in 1986 by that great writer of popular science and science fiction Isaac Asimov: 'Inspect every piece of pseudoscience and you will find a security blanket, a thumb to suck, a skirt to hold. ' Childhood is, for many people, a lost Arcadia, a kind of heaven, with its certainties and its securities, its fantasies of flying to the Never Never Land, its bedtime stories before we drifted off to the Land of Nod in the arms of Teddy Bear. With hindsight, the years of childish innocence may pass too soon. I love my parents for taking me for a ride, high as a kite,
through the treetops; and for telling me about the Tooth Fairy and Father Christmas, about Merlin and his spells, about baby Jesus and the Three Wise Men, All these stories enrich childhood and, together with so many other things, help to make it, in memory, a time of enchantment.
The adult world may seem a cold and empty place, with no fairies and no Father Christmas, no Toyland or Narnia, no Happy Hunting Ground where mourned pets go, and no angels - guardian or garden variety. But there are also no devils, no hellfire, no wicked witches, no ghosts, no haunted houses, no daemonic possession, no bogeymen or ogres. Yes, Teddy and Dolly turn out not to be really alive. But there are warm, live, speaking, thinking, adult bedfellows to hold, and many of us find it a more rewarding kind of love than the childish affection for stuffed toys, however soft and cuddly they may be.
Not to grow up properly is to retain our 'caterpillar' quality from childhood (where it is a virtue) into adulthood (where it becomes a vice). In childhood our credulity serves us well. It helps us to pack, with extraordinary rapidity, our skulls full of the wisdom of our parents and our ancestors. But if we don't grow out of it in the fullness of time, our caterpillar nature makes us a sitting target for astrologers, mediums, gurus, evangelists and quacks. The genius of the human child, mental caterpillar extraordinary, is for soaking up information and ideas, not for criticizing them. If critical faculties later grow it will be in spite of, not because of, the inclinations of childhood. The blotting paper of the child's brain is the unpromising seedbed, the base upon which later the sceptical attitude, like a struggling mustard plant, may possibly grow. We need to replace the automatic credulity of childhood with the constructive scepticism of adult science.
But I suspect an additional problem. Our story of the child as
information caterpillar was too simple. The programming of the child's credulity has a twist which, until we understand it, is almost paradoxical. Let us go back to our picture of the child needing to absorb information from the previous generation as swiftly as possible. What if two adults, say your mother and your father, give you contradictory advice? What if your mother tells you that all snakes are deadly and you must never go near them, but next day your father tells you that all snakes are deadly except green ones and you can keep a green snake as a pet? Both pieces of advice may be good. The mother's more general advice has the desired effect of protecting you against snakes, even though it is too sweeping when it comes to green snakes. The father's more discriminating advice has the same protective effect and is in some ways better, But it could be fatal if carried, unrevised, to a distant country. In any case, to the young child the contradiction between the two might be dangerously confusing. Parents often make strenuous efforts not to contradict one another, and
they are probably wise to do so. But natural selection, in 'designing' credulity, would need to build in a way of coping with contradictory advice. Perhaps a simple override rule, such as 'Believe whichever story you heard first. ' Or 'Believe mother rather than father, and father rather than other adults in the population. '
Sometimes the advice from parents is specifically aimed against credulity towards other adults in the population. The following is a piece of advice that parents need to give their children: 'If any adult asks you to come with him and says that he is a friend of your parents, don't believe him, however nice he seems and even (or especially) if he offers you sweets. Only go with an adult that you and your parents already know, or who is wearing a policeman's uniform. ' (A charming story recently appeared in the English newspapers in which Queen Elizabeth the Queen Mother, aged 97, told her chauffeur to stop the car when she noticed a crying child who was apparently lost. The kind old lady got out to comfort the little girl and offered to take her home. 'I can't,' wailed the child, 'I'm not allowed to talk to strangers. ') A child is called upon to exercise the exact opposite of credulity in some circumstances: a tenacious persistence in believing an earlier adult statement in the face of what may be a tempting and plausible - but contradictory - later statement.
On their own, then, the words 'gullible' and 'credulous' are not quite right for children. Truly credulous people believe whatever they have most recently been told, even if this contradicts what others have told them before. The quality of childhood that I am trying to pin down is not pure gullibility but a complex combination of gullibility coupled with its opposite - stubborn persistence in a belief, once acquired. The full recipe, then, is extreme early gullibility followed by equally obstinate subsequent unshakeability. You can see what a devastating combination this could be. Those old Jesuits knew what they were about: 'Give me the child for his first seven years, and I'll give you the man. '
7
UNWEAVING THE UNCANNY
. . . though no great ministering reason sorts Out the dark mysteries of human souls
To clear conceiving. . .
JOHN KEATS, 'Sleep and Poetry' (1817)
The eminent fertility specialist Robert Winston imagines the following advertisement, placed in the newspaper by an unscrupulous quack
doctor, aimed at people who want their next baby to be, say, a son (the sexism underlying this assumption is not mine but could be found unquestioned all over the ancient world, and still in many places today). 'Send ? 500 for my patent recipe to make your baby a boy. Money refunded in full if I fail. ' The money back guarantee is intended to establish confidence in the method. In fact, of course, since boys turn up anyway on approximately 50 per cent of occasions, the scheme would be a nice little earner. Indeed, the quack could safely offer compensation of, say, ? 250 for every girl born, over and above the money back guarantee. He would still show a tidy profit in the long run.
I used a similar illustration in one of my Royal Institution Christmas Lectures in 1991. I said I had reason to believe that among my audience was a psychic, clairvoyant individual, capable of influencing events purely by the power of thought. I would try to flush this individual out. 'Let's first establish,' I said, 'whether the psychic is in the left half or the right half of the lecture hall. ' I invited everybody to stand up while my assistant tossed a coin.
Everybody on the left of the hall was asked to 'will' the coin to come down heads. Everybody on the right had to will it to be tails. Obviously one side had to lose, and they were asked to sit down. Then those that remained were divided into two, with half 'willing' heads and the other half tails. Again the losers sat down. And so on by successive halvings until, inevitably, after seven or eight tosses, one individual was left standing. 'A big round of applause for our psychic' He must be psychic, mustn't he, because he successfully influenced the coin eight times in a row?
If the lectures had been televised live, instead of recorded and broadcast later, the demonstration would have been much more impressive. I'd have asked everybody who watched it whose surname begins before J in the alphabet to 'will' heads and the rest tails. Whichever half turned out to contain the 'psychic' would have been divided in half again, and so on. I'd have asked everybody to keep a written record of the order of their 'willings'. With two million viewers, it would have taken about 21 steps to narrow down to a single individual. To be on the safe side I'd have stopped a bit short of 21 steps. At, say, the eighteenth step I'd have invited anybody still in the game to phone in. There would have been quite a few and, with luck, one would phone. This individual would then have been invited to read out his/her written record: which would have matched the official record. So this one individual succeeded in influencing 18 successive tosses of a coin. Gasps of admiration. But admiration for what? Nothing but pure luck. I don't know if that experiment has been done. Actually, the trick here is so obvious it probably wouldn't fool many people. But how about the following?
A well-known 'psychic' goes on television, a lucrative engagement fixed up over lunch by his publicity agent. Staring out of ten million screens with hypnotically smouldering eyes (nice job by Make-up and Lighting), our imaginary seer intones that he feels a strange, spiritual rapport, a vibrating resonance of cosmic energy, with certain members of his audience. They will be able to tell who they are because, even as he utters his mystic incantation, their watches will stop. After only a brief pause, a telephone on his table rings and an amplified voice in awed tones announces that its owner's watch stopped dead within seconds of the clairvoyant's words. The caller adds that she had a premonition that this was going to happen even before she looked down at her watch, for something in her hero's burning eyes seemed to speak directly to her soul. She felt the 'vibrations' of 'energy'. Even as she is speaking, a second telephone rings. Yet another watch has stopped.
A third caller's grandfather clock stopped - surely a weightier feat than stopping a little watch whose delicate hairspring would naturally be more susceptible to psychic forces than the massive pendulum of the grandfather! Another viewer's watch actually stopped a little before the celebrated mystic made his pronouncement - is this not an even more impressive feat of psychic control? Yet another watch has been more impatiently susceptible to occult forces. It had stopped a whole day before, at the very moment when its owner looked at the famous mystic's photograph in the newspaper. The studio audience gasps its appreciation. This, surely, is psychic power beyond all scepticism, for it happened a whole day early! 'There are more things in heaven and earth, Horatio . . . '
What we need is less gasping and more thinking. This chapter is about how to take the sting out of coincidence by quietly sitting down and calculating the likelihood that it would have happened anyway. In the course of this, we shall discover that to disarm apparently uncanny coincidences is more interesting than gasping over them anyway.
Sometimes the calculation is easy. In a previous book I gave away the number of the combination lock on my bicycle. I felt safe in doing so because obviously my books would never be read by the kind of person who would steal a bicycle. Unfortunately somebody did steal it, and I now have a new lock with a new number, 4167. I find this number easy to remember. 41 is imprinted in my memory as the arbitrary code used to identify my clothes and shoes at boarding school. 67 is the age at which I am due to retire. Obviously there is no interesting coincidence here: whatever the number had been, I'd have searched my life for a mnemonic recipe and I'd have found it. But mark the sequel. On the day of writing this, I received from my Oxford college a letter saying:
Each person authorized to use the photocopiers is issued with a personal code number which permits access. Your new number is 4167.
My first thought was that I'd undoubtedly lose this piece of paper (I quickly lost its equivalent last year) and I must immediately think of a formula to fix it in my memory. Something similar to the mnemonic by which I remember my bicycle combination, perhaps? So I looked again at the number on the letter and, to borrow a neat line from Fred Hoyle's science fiction novel The Black Cloud, the figures on the piece of paper seemed to swell to a gigantic size. I didn't need a new mnemonic. The number was identical. I rushed to tell my wife of the amazing coincidence, but on more sober reflection I shouldn't have bothered.
The odds of this happening by chance alone are easily calculated. The first digit could have been anything from 0 to 9. So there is a one in 10 chance of getting a 4 and matching the bicycle lock. For each of these ten possibilities, the second digit could have been anything from 0 to 9, so again there is a one in 10 chance of matching the bike lock's second dial. The odds of matching the first two digits is therefore one in 100 and, following the logic through the other two digits, the odds of matching all four digits of the bicycle lock is one in 10,000. It is this large number that is our protection against theft.
The coincidence is impressive. But what should we conclude? Has something mysterious and providential been going on? Have guardian angels been at work behind the scenes? Have lucky stars swum into Uranus? No. There is no reason to suspect anything more than simple accident. The number of people in the world is so large compared with 10,000 that somebody, at this very moment, is bound to be experiencing a coincidence at least as startling as mine. It just happens that today was my day to notice such a coincidence. It isn't even an added coincidence that it happened to me on this particular day, while I was writing this chapter. I had in fact written the first draft of the chapter some weeks ago. I reopened it today, after the coincidence occurred, in order to insert this anecdote. I shall surely reopen it many times to revise and polish, and I shall not remove the references to 'today': they were accurate when written. This is another way in which we habitually inflate the impressiveness of coincidence in order to make a good story.
We can do a similar calculation for the television guru whose psychic miasma seemed to stop people's watches, but we'll have to use estimates rather than exact figures. Any given watch has a certain low probability of stopping at any moment. I don't know what this probability is, but here's the kind of way in which we could come to an estimate. If we take just digital watches, their battery typically runs out within a year. Approximately, then, a digital watch stops once per year. Presumably clockwork watches stop more often because people forget to wind them
and presumably digital watches stop less often because people sometimes remember to renew the battery ahead of time. But both kinds of watches probably stop as often again because they develop faults of one kind or another. So, let our estimate be that any given watch is likely to stop about once a year. It doesn't matter too much how accurate our estimate is. The principle will remain.
If somebody's watch stopped three weeks after the spell was cast, even the most credulous would prefer to put it down to chance. We need to decide how large a delay would have been judged by the audience as sufficiently simultaneous with the psychic's announcement to impress. About five minutes is certainly safe, especially since he can keep talking to each caller for a few minutes before the next call ceases to seem roughly simultaneous. There are about 100,000 five-minute periods in a year. The probability that any given watch, say mine, will stop in a designated five-minute period is about 1 in 100,000.
Low odds, but there are 10 million people watching the show. If only half of them are wearing watches, we could expect about 25 of those watches to stop in any given minute. If only a quarter of these ring in to the studio, that is 6 calls, more than enough to dumbfound a naive audience. Especially when you add in the calls from people whose watches stopped the day before, people whose watches didn't stop but whose grandfather clocks did, people who died of heart attacks and their bereaved relatives phoned in to say that their 'ticker' gave out, and so on. This kind of coincidence is celebrated in the delightfully sentimental old song, 'Grandfather's Clock:'
Ninety years without slumbering. Tick, lock, tick, tock.
His life seconds numbering. Tick, tock, tick, tock.
It stopped . . . short. . . never to go again When the old man died.
Richard Feynman, in a 1963 lecture published posthumously in 1998, tells the story of how his first wife died at 9. 22 in the evening and the clock in her room was later found to have stopped at exactly 9. 22. There are those who would revel in the apparent mystery of this coincidence and feel that Feynman has taken away something precious when he gives us a simple, rational explanation of the mystery. The clock was old and erratic and was in the habit of stopping if tilted out of the horizontal. Feynman himself frequently repaired it. When Mrs Feynman died, the
nurse's duty was to record the exact time of death. She moved over to the clock, but it was in dark shadow. In order to see it, she picked it up - and tilted its face towards the light . . . The clock stopped. Is Feynman really spoiling something beautiful when he tells us what is surely the true - and very simple - explanation? Not for my money. For me, he is affirming the elegance and beauty of an orderly universe in which clocks stop for reasons, not to titillate human sentimental fancy.
At this point, I want to invent a technical term, and I hope you'll forgive an acronym. PETWHAC stands for Population of Events That Would Have Appeared Coincidental. Population may seem an odd word, but it is the correct statistical term. I won't keep using capital letters because they stand so unattractively on the page. Somebody's watch stopping within ten seconds of the psychic's incantation obviously belongs within the petwhac, but so do many other events. Strictly speaking, the grandfather clock's stopping should not be included. The mystic did not claim that he could stop grandfather clocks. Yet when somebody's grandfather clock did stop, they immediately telephoned in because they were, if anything, even more impressed than they would have been if their watch had stopped. The odd misconception is fostered that the psychic is even more powerful since he didn't even bother to mention that he could stop grandfather clocks, too! Similarly, he said nothing about watches stopping the day before or grandfathers' tickers suffering cardiac arrests.
People feel that such unanticipated events belong in the petwhac. It looks to them as though occult forces must have been at work. But when you start to think like this, the petwhac becomes really quite large, and therein lies the catch. If your watch stopped exactly 24 hours earlier, you would not have to be unduly gullible to embrace this event within the petwhac. If somebody's watch stopped exactly seven minutes before the spell, this might impress some people because seven is an ancient mystic number. And the same presumably goes for seven hours, seven days . . . The larger the petwhac, the less we ought to be impressed by the coincidence when it comes. One of the devices of an effective trickster is to make people think exactly the opposite.
By the way, I deliberately chose a more impressive trick for my imaginary psychic than is actually done with watches on television. The more familiar feat is to start watches that have stopped. The television
audience is invited to get up and fetch, out of drawers or attics, watches that have broken down, and hold them while the psychic performs some incantation or does some hypnotic eye work. What is really going on is that the warmth of the hand melts oil that has coagulated and this starts the watch ticking, if only briefly. Even if this happens in only a small proportion of cases, this proportion, multiplied by the large audience, will generate a satisfactory number of dumbfounded telephone calls. Actually,
as Nicholas Humphrey explains in his admirable expose of supernaturalism Soul Searching (1995), it has been demonstrated that more than 50 per cent of broken watches start, at least momentarily, if they are held in the hand.
Here's another example of a coincidence, where it is clear how to calculate the odds. We shall use it to go on and see how odds are sensitive to changing the petwhac. I once had a girlfriend who had the same birth date (though not in the same year) as my previous girlfriend. She told a friend of hers who believed in astrology, and the friend triumphantly asked how I could possibly justify my scepticism in the face of such overwhelming evidence that I had unwittingly been brought together with two successive women on the basis of their 'stars'. Once again, let's just think it through quietly. It is easy to calculate the odds that two people, chosen entirely at random, will have the same birthday. There are 565 days in the year. Whatever the birthday of the first person, the chance that the second will have the same birthday is 1 in 365 (forgetting leap years). If we pair people off in any particular way, such as taking the successive women friends of any one man, the odds that they will share their birthday are 1 in 365. If we take ten million men (less than the population of Tokyo or Mexico City), this apparently uncanny coincidence will have happened to more than 27,000 of them!
Now let's think about the petwhac and see how the apparent coincidence becomes less impressive as it swells. There are many other ways in which we could pair people off and still end up noticing an apparent coincidence. Two successive girlfriends with the same surname, although unrelated, for instance. Two business partners with the same birthday would also come within the petwhac; or two people with the same birthday sitting next to one another on an aeroplane. Yet, in a well- loaded Boeing 747, the odds are actually better than 50 per cent that at least one pair of neighbours will share a birthday. We don't usually notice this because we don't look over each other's shoulders as we fill in those tedious immigration forms. But if we did, somebody on most flights would go away muttering darkly about occult forces.
The birthday coincidence is famously phrased in a more dramatic way. If you have a roomful of only 25 people, mathematicians can prove that the odds are just greater than 50 per cent that at least two of them will share the same birthday. Two readers of an earlier draft of the book asked me to justify this astonishing statement. It's easier to calculate the odds that there won't be any shared birthdays and subtract from one. Forget about leap years because they're more trouble than they're worth. Suppose I bet you that with 25 people in a room, at least two will share a birthday. You bet, for the sake of argument, that there'll be no shared birthdays. We're going to do the calculation by working up to 23 people gradually,
starting with just one person in the room, and adding people one at a time. If at any point a match is found, I've won the bet, we stop the game and don't bother to add any more people. If we get to 23 people and there's still no match so far, you win the bet.
When the room contains only the first person, whom we may as well call A, the chance of 'no match so far' is trivially 1 (365 out of 365 chances). Now add a second person, B. The chance of a match is now one in 365. So the odds of 'no match so far' when B has joined A in the room are 364/365. Now add a third person, C, There's a one in 365 chance that C matches A and a one in 365 chance that C matches B, therefore the chance that he matches neither A nor B is 363/365 (he can't match both, because we already know that A doesn't match B). To get the total odds
of 'no match so far' we have to take this 363/365 and multiply it by the odds against a match in the previous round (s), in this case by 364/365. The same reasoning applies when we add the fourth person, D. The total odds of 'no match so far' are now 364/365 X 363/365 X 362/365. And so on until all 23 people are in the room. Each new person adds a new term that we have to bring in to the running multiplication sum, to compute 'no match so far'.
If you multiply this out for 23 terms (you have to go on down to 343/365) the answer comes to about 0. 49. This is the chance that there will not be any shared birthdays in the room. So there's a slightly greater than even chance that at least one pair of individuals in a committee of 23 will share a birthday. Most people's intuition would encourage them to bet against such a coincidence. But they'd be wrong. It is this kind of intuitive error that in general bedevils our assessment of 'uncanny' coincidences.
Here's an actual coincidence where, although it is a little harder, we can make a stab at estimating the odds approximately. My wife once bought for her mother a beautiful antique watch with a pink face. When she got it home and peeled off the price label she was amazed to find, engraved on the back of the watch, her mother's own initials, M. A. B. Uncanny? Eerie? Spine-crawling? Arthur Koestler, the famous novelist, would have read much into it. So would C. G. Jung, the widely admired psychologist and inventor of the 'collective unconscious', who also believed that a bookcase or a knife might be induced by psychic forces to explode spontaneously with a loud report. My wife, who has more sense, merely thought the coincidence of initials remarkably convenient and sufficiently amusing to justify telling the story to me - and here I am now telling it to a wider audience.
So, what really are the odds against a coincidence of this magnitude? We can begin by calculating them in a naive way. There are 26 letters in the
alphabet. If your mother has three initials and you find a watch engraved with three letters at random, the odds that the two will coincide is 1/26 x 1/26 x 1/26, or one in 17,576. There are about 55 million people in Britain. If every one of them bought an antique engraved watch we'd expect more than 3,000 of them to gasp with amazement when they discovered that the watch already bore their mother's initials.
But the odds are actually better than this. Our naive calculation made the incorrect assumption that each letter has a probability of 1/26 of being somebody's initial. This is the average probability for the alphabet as a whole, but some letters, such as X and Z, have a smaller probability. Others, including M, A and B, are commoner: think how much more impressed we'd be if the coinciding initials had been X. Q. Z. We can improve our estimate of odds by sampling a telephone directory. Sampling is a respectable way of estimating something that we cannot count directly. The London directory is a good place to sample because it is large and London happens to be where my wife bought the watch and where her mother lived. The London telephone directory contains about 85,060 column inches, or about 1. 34 column miles, of private citizens' names. Of these, about 8,110 column inches are devoted to the Bs. This means that about 9. 5 per cent of Londoners have a surname beginning with B - much more frequent than the figure for an average letter: 1/26, or 3. 8 per cent.
So, the probability that a randomly chosen Londoner would have a surname beginning with B is about 0. 095 (~ 9. 5 per cent). What about
the corresponding probabilities that the forenames will begin with M or A? It would take too long to count forename initials right through the telephone book, and there'd be no point since the telephone book is itself only a sample. The easiest thing to do is take a subsample where forename initials are conveniently arranged in alphabetical order. This is true of the listings within any one surname. I shall take the commonest surname in England - Smith - and look at what proportion of the Smiths are M. Smith and what proportion are A. Smith. It is a reasonable hope that this will be approximately representative of the probabilities of forename initials for Londoners generally. It turns out that there are rather more than 20 column yards of Smiths altogether. Of these, 0. 075 them (53. 6 column inches) are M. Smiths. The A. Smiths fill 75-4 column inches, representing 0. 102 of all the Smiths,
If you are a Londoner and you have three initials, therefore, the chances of your initials being M. A. B. in that order are approximately 0. 102 X 0. 073 0-095 or about 0. 0007. Since the population of Britain is 55 million, this should mean that about 38,000 of them have the initials M. A. B. , but only if everybody among those 55 million has three initials. Obviously not everybody does but, looking down the telephone directory
again, it seems that at least a majority do. If we make the conservative assumption that only half of British people have three initials, that still means that more than 19,000 British people have identical initials to my wife's mother. Any one of them could have bought that watch and gasped with astonishment at the coincidence. Our calculation has shown that there is no reason to gasp.
Indeed, when we think harder about the petwhac, we find that we have even less right to be impressed. M. A. B. were the initial letters of my wife's mother's maiden name. Her married initials of M. A. W. would have seemed just as impressive had they been found on the watch. Surnames beginning with W are nearly as common in the telephone book as those beginning with B. This consideration approximately doubles the petwhac, by doubling the number of people in the country who would have been deemed, by a coincidence hunter, to have 'the same initials' as my wife's mother.
Moreover, if somebody bought a watch and found it to be engraved not with her mother's initials but with her own, she might consider it an even greater coincidence and more worthy to be embraced within the (ever-growing) petwhac.
The late Arthur Koestler, as I have already mentioned, was a great enthusiast of coincidences. Among the stories that he recounts in The Roots of Coincidence (1972) are several originally collected by his hero, the Austrian biologist Paul Kammerer (famous for publishing a faked experiment purportedly demonstrating the 'inheritance of acquired characteristics' in the midwife toad). Here is a typical Kammerer story quoted by Koestler:
On September 18, 1916, my wife, while waiting for her turn in the consulting rooms of Prof. Dr J. v. H. , reads the magazine Die Kunst; she is impressed by some reproductions of pictures by a painter named Schwalbach, and makes a mental note to remember his name because she would like to see the originals. At that moment the door opens and the receptionist calls out to the patients: 'Is Frau Schwalbach here? She is wanted on the telephone. '
It probably isn't worth trying to estimate the odds against this coincidence, but we can at least write down some of the things that we'd need to know. 'At that moment the door opens' is a little vague. Did the door open one second after she made the mental note to look up Schwalbach's paintings or 20 minutes? How long could the interval have been, leaving her still impressed by the coincidence? The frequency of the name Schwalbach is obviously relevant: we'd be less impressed if it had been Schmidt or Strauss; more impressed if it had been Twistleton- Wykeham-Fiennes or Knatchbull-Huguesson. My local library doesn't
have the Vienna telephone book, but a quick look in another large Germanic telephone directory, the Berlin one, yields half a dozen Schwalbachs: the name is not particularly common, therefore, and it is understandable that the lady was impressed. But we need to think further about the size of the petwhac. Similar coincidences could have happened to people in other doctors' waiting rooms; and in dentists' waiting rooms, government offices and so on; and not just in Vienna but anywhere else. The quantity to keep bearing in mind is the number of opportunities for coincidence that would have been thought, if they had occurred, just as remarkable as the one that actually did occur.
Now let's take another kind of coincidence, where it is even harder to know how to start calculating odds. Consider the often-quoted experience of dreaming of an old acquaintance for the first time in years and then getting a letter from him, out of the blue, the next day. Or of learning that he died in the night. Or of learning that he didn't die in the night but his father did.
