Indeed, coincidence may be defined as the very tool used by Fate to shape the destinies of men and nations.

Rafael Sabatini
Captain Blood, Dover, 2004, p. 136

In a group of 23 people, at least two have the same birthday with the probability higher than 1/2.

This could be reformulated the following way. There is a bag of 365 different items. We pick an item at random. Note which one it was and return it back into the bag. The question: how many items should we pick so that with a probability higher than 1/2 we would have picked the same item twice. The items in the bag are assumed to be thoroughly mixed before every trial. Is it not surprising that all it takes is 23 items?

To prove our assertion let us start with just two people. What is the probability p2 that two random persons have the same birthday? It's easier to answer a related question. What is the probability q2 that two random persons have different birthdays? Obviously, p2 + q2 = 1. Thus answering one question we automatically get an answer to another.

Thus we have a person with a birthday which falls onto one of the 365 days and ask what is the probability that another person has a different birthday. Since any day out of 365 but the birthday of the first person would make a different birthday, q2 = 364/365.

Consider now three people. What is the probability q3 that no two of them have the same birthday? Obviously p3 + q3 = 1, where p3 is the probability that at least two of the group have the same birthday. As before, take one fellow and his birthday. The second has now 364 days to choose from and, if the third was born on any of the remaining 363 days they would form a "no-overlapping-birthday" group. Thus q3 = 364/365·363/365.

Proceeding in this fashion we'll get q4 = (364/365)·(363/365)·(362/365), and so on. Since every fraction in these products is less than 1, the sequence q2, q3, q4, ... is decreasing. Therefore, the sequence p2, p3, p4, ... is increasing. Now, perhaps, it would be less surprising to learn that p23 > 1/2. Recollect that

q23 = 364/365 · 363/365 · ... · 343/365

(This probability can be An Exercise without Calculating Aids">computed manually with a relatively little effort.)

Here is a simpler derivation (R. Pinkham, The Mathematical Gazette, v 69, n 450 (Dec 1985), 279).

23 people can be combined into 23·22/2 = 253 pairs. The probability that in any one pair the fellows have the same birthday is 1/365. 253 tries each with probability 1/365 of success add up to 253/365 > 1/2.

This derivation is faulty. Do you see why? (Hint: try the same calculation with 22 people.)

The Theory of Probability supplies not a small number of unexpected results, controversial problems and paradoxes.

Following is an excerpt from
Beyond Numeracy

Coincidences fascinate us. They seem to compel a search for their significance. More often than some people realize, however, they're to be expected and require no special explanation. Surely no cosmic conclusions may be drawn from the fact that I recently and quite by accident met someone in Seattle whose father had played on the same Chicago high school baseball team as my father had and whose daughter is the same age and has the same name as my daughter. As improbable as this particular event was (or as particular events always are), that some event of this vaguely characterized sort should occasionally occur is very likely.

More precisely, it can be shown, for example, that if two strangers sit next to each other on an airplane, more than 99 times out of 100 they will be linked in some way by two or fewer intermediates. (The linkage with my father's classmate was more striking. It was via only one intermediate, my father, and contained other elements.) Maybe, for example, the cousin of one of the passengers will know the other's dentist. Most of the time people won't discover these links, since in casual conversation they don't usually run through all their 1,500 or so acquaintances as well as all their acquaintances' acquaintances. (I suppose with laptop computers becoming more popular they could compare their own personal databases and even those of people they know. Perhaps exchanging databases might soon be as common as leaving a business card. Electronic networking. Hellacious.)

There is a tendency, however, to home in on likely co-acquaintances. Such connections are thus discovered frequently enough so that the squeals of amazement that commonly accompany their discovery are unwarranted. Similarly unimpressive is the "prophetic" dream which traditionally comes to light after some natural disaster has occurred. Given the half billion hours of dreaming each night in this country - 2 hours per night for 200 million people - we should expect as much.

Or consider the famous birthday problem in probability theory. One must gather together 367 people (one more than the number of days in a leap year) in order to ensure that 2 of them share a birthday. But if one is willing to settle for a 50-50 chance of this happening, only 23 people need be gathered. Rephrasing, I note that if we imagine a school with thousands of classrooms each of which contains 23 students, then approximately half of these classrooms will contain 2 students who share a birthday. No time should be wasted trying to explain the meaning of these or other coincidences of similar type. They just happen.

One somewhat different example concerns the publisher of a stock newsletter who sends out 64,000 letters extolling his state-of-the-art database, his inside contacts, and his sophisticated econometric models. In 32,000 of these letters he predicts a rise in some stock index for the following week, say, and in 32,000 of them he predicts a decline. Whatever happens, he sends a follow-up letter but only to those 32,000 to whom he's made a correct "prediction." To 16,000 of them he predicts a rise for the next week, and to 16,000 a decline. Again, whatever happens, he will have sent 2 consecutive correct predictions to 16,000 people. Iterating this procedure of focusing exclusively on the winnowed list of people who have received only correct predictions, he can create the illusion in them that he knows what he's talking about. After all, the 1,000 or so remaining people who have received 6 straight correct predictions (by coincidence) have a good reason to cough up the $1,000 the newsletter, publisher requests: They want to continue to receive these "oracular" pronouncements.

I repeat that a useful distinction in discussing these and other coincidences is that between generic sorts of events and particular events. Many situations are such that the particular event that occurs is guaranteed to be rare - a certain individual winning the lottery or a specific bridge hand being dealt - while the generic outcome - someone's winning the lottery or some bridge hand being dealt-is unremarkable. Consider the birthday problem again. If all that we require is that 2 people have some birthday in common rather than any particular birthday, then 23 people suffice to make this happen with a probability of 1/2. By contrast, 253 people are needed in order for the probability to be 1/2 that one of them has a specific birth date, say July 4. Particular events specified beforehand are, of course, quite difficult to forecast, so it's not surprising that predictions by televangelists, quack doctors, and others are usually vague and amorphous (that is, until the events in question have occurred, at which time the prognosticators like to assert that these precise outcomes were indeed foreseen).

This brings me to the so-called Jeane Dixon effect, whereby the few correct predictions (by psychics, disreputable stock newsletters, whomever) are widely heralded and the 9,839 or so false predictions made annually are conveniently ignored. The phenomenon is quite widespread and contributes to the tendency we all have to read more significance into coincidences than is usually justified. We forget all the premonitions of disaster we've had which didn't predict the future and remember vividly those few which seemed to do so. Instances of seemingly telepathic thought are reported to everyone we know; the incomparably vaster number of times this does not occur are too banal to mention.

Even our biology conspires to make coincidences appear more meaningful than they usually are. Since the natural world of rocks, plants, and rivers does not seem to offer much evidence for superfluous coincidences, primitive man had to be very sensitive to every conceivable anomaly and improbability as he slowly developed science and its progenitor, "common sense." Coincidences, after all, are sometimes quite significant. In our complicated and largely man-made modem world, however, the plethora of connections among us appears to have overstimulated many people's inborn tendency to note coincidence and improbability and led them to postulate causes and forces where there are none. People know more names (not only family members' but also those of colleagues and a myriad of public figures), dates (from news stories to personal appointments and schedules), addresses (whether actual physical ones or telephone numbers, office numbers, and so on), and organizations and acronyms (from the FBI to the IMF, from AIDS to ASEAN) than ever before. Thus, although it is a very difficult quantity to measure, the rate at which coincidences occur has probably risen over the last century or two. Still, for most of them it generally makes little sense to demand an explanation.

In reality, the most astonishingly incredible coincidence imaginable would be the complete absence of all coincidences. Then again, some things that appear coincidental are not random at all: say, among 100 people gathered in a room, some are (mutually) friends and some are not. What is the probability that two people have (among the present) the same number of friends?


  1. J. Havil, Nonplussed!, Princeton University Press, 2007

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