Coincidences: remarkable or random? - What are the Chances? - Cover Story

Most improbable coincidences likely result from play of random events. The very nature of randomness assures that combing random data will yield some pattern.

"You don't believe in telepathy?" My friend, a sober professional, looked askance. "Do you?" I replied. "Of course. So many times I've been out for the evening and suddenly became worried about the kids. Upon calling home, I've learned one is sick, hurt himself, or having nightmares. How else can you explain it?"

Such episodes have happened to us all and it's common to hear the words, "It couldn't be just coincidence." Today the explanation many people reach for involves mental telepathy or psychic stirrings. But should we leap so readily into the arms of a mystic realm? Could such events result from coincidence after all?

There are two features of coincidences not well known among the public. First, we tend to overlook the powerful reinforcement of coincidences, both waking and in dreams, in our memories. Non-coincidental events do not register in our memories with nearly the same intensity. Second, we fail to realize the extent to which highly improbable events occur daily to everyone. It is not possible to estimate all the probabilities of many paired events that occur in our daily lives. We often tend to assign coincidences a lesser probability than they deserve.

However, it is possible to calculate the probabilities of some seemingly improbable events with precision. These examples provide clues as to how our expectations fail to agree with reality.

Coincident Birthdates

In a random selection of twenty-three persons there is a 50 percent chance that at least two of them celebrate the same birthdate. Who has not been surprised at learning this for the first time? The calculation is straightforward. First find the probability that everyone in a group of people have different birthdates (X) and then subtract this fraction from one to obtain the probability of at least one common birthdate in the group (P), P = 1 - X. Probabilities range from 0 to 1, or may be expressed as 0 to 100%. For no coincident birthdates a second person has a choice of 364 days, a third person 363 days, and the nth person 366 - n days. So the probability for all different birthdates becomes

For two people: [X.sub.2] = 365/365 [multiplied by] 364/365

For three people: [X.sub.3] = 365/365 [multiplied by] 364/365 [multiplied by] 363/365

For n people: Xn = 365/365 [multiplied by] 364/365 [multiplied by] 363/365 ... 366-n/365 = 365!/(365-n)! 365"

With its factorials the last equality is not especially useful unless one possesses the capability of handling very large numbers. It is instructive to use a spreadsheet or a loop in a computer language to calculate Xn from the first equality for successive values of n. When n = 23, one finds X = 0.493 and P = 0.507. A plot of the probability of at least one common birthdate, P, versus the number of people, n, appears as the right hand curve of circles in Figure 1. The curve shows that the probability of at least two people sharing a common birthdate rises slowly, at first passing just less than 12% probability with ten people, rising through 50% probability at the open circle corresponding to twenty-three people, then flattening out and reaching 90% probability in a group of forty-one people. This means that on the average, out often random groups of forty-one persons, in nine of them at least two persons will celebrate identical birthdates. No mysterious forces are needed to explain this coincidence.

Note that the probability of coincident birthdays for 2x23 = 46 people is not 100%, as some might suppose, but 95% as shown by the right-hand curve in Figure 1. Extension of the curve beyond the limit of Figure 1 reveals that fifty-seven people produce a 99% probability of coincident birthdays.

The same principle may be used to calculate the probability that at least two people in a random group possess birthdates within one day (same and two adjacent days). This condition is less restrictive than the former, and 50% probability is passed with just fourteen people. The left-hand curve in Figure 1 shows a plot for the probabilities of within-one-day birthdates.

Delving a little deeper into some aspects of the probabilities of identical birthdates provides additional insight. Note that we said several times "at least two people" sharing a common birthdate. As the group size increases the chances for multiple coincidences also increase. The descending curve at the left of Figure 2 represents the probability of no coincidences (NC) of birthdates, identical to the Xn values calculated above. The first curve with a maximum plots the probability of only one pair (1P) sharing an identical birthdate. The maximum occurs at twenty-eight people with a probability of almost 0.39. As the group becomes larger the probability of other coincidences increases as well. The second curve with a maximum represents the probability of exactly two pairs (2P) sharing an identical birthdate. Its maximum occurs at thirty-nine people with a probability of 0.28. The last, rising curve in Figure 2 plots the total probabilities of all remaining coincidences ([greater than]2P), consisting of three pairs, triplets, etc. For all numbers of people, the probabilities of all four curves total 1.00.