“When a coincidence seems amazing, that’s because the human mind isn’t wired to naturally comprehend probability and statistics.” [Neil deGrasse Tyson]
The unreasonable popularity of pseudosciences such as ESP or astrology often stems from personal experience. We’ve all had that “Ralph” phone call or some other happening that seems well beyond the range of normal probability, at least according to what we consider to be common sense. But how accurately does common sense forecast probabilities and how much of it is fuzzy math? As we will see, fuzzy math holds its own.
Let’s try to de-fuzz the math a bit, starting with a classic example: the birthday problem. Perhaps you’ve encountered this problem in a math class that dealt with probabilities. In a group of people, what is the probability that two people would have the same birthday? Certainly it must depend on the size of the group. If we start with only two people, the chance would be one out of 365 – well, OK, one in 366 for leap years. If the group included more people, common sense might suggest that the probability would just increase linearly. So, to get a 50% chance, you might think it would take 183 people in the group. Wrong. That’s where common sense goes off the rails. It turns out that, in a group of only 23 people, the probability of two having the same birthday is 50%.
Details of the logic required to arrive at this result are unnecessary here, but a clue is given by a group of three. The third person might match the birthday of either of the first two, so you might think to just double the first probability. But think about this from the inverse point of view. The probability of the second person’s birthday NOT matching the first is 364/365. But the third person could match either of the first two, so the probability of NOT matching is only 363/365. Since NOT matching is thus less probable, matching becomes more probable. Working this out involves a bit of number crunching, but math classes have calculators galore, and since many classes have 23 or more members, real data are available to support the probability calculation. As you can see, what we take to be common sense often yields inaccurate solutions.
Meanwhile, back at the “Ralph” problem, a math textbook might tackle this problem in terms of drawing different colored pebbles from a large urn. Let’s forego that approach, and set the “Ralph” problem in more realistic terms. Suppose you know N people. During the course of a single day, a number of those people, k, cross your mind on a purely random basis. For this illustration, let’s agree to ignore close relatives and friends that you think about almost every day. Next, a certain number of people, L, contact you in a given day by any means, including phone calls, e-mails and electronic messages, social media, snail mail, and random meetings.
Working though this problem (actually kind of fun if you like mathematical puzzles) yields an equation for the number of days that will elapse before the probability of getting a contact from someone you thought about reaches a given level. Of course, it depends on the variables N, k, and L, not the easiest quantities to obtain.
An estimate of N, the number of people that an average person knows, is available from various sources, and ranges from 200 to 1500, but k, the number of people one would think about is highly subjective, as is L, the number of contacts one receives in an average day. Yet, all these numbers are necessary to find an estimate of the time required for someone you thought about to contact you shortly after you thought about them. Unscientific surveys of students, neighbors, and friends produced numbers of thoughts from 10 to 100 and contacts from 5 to 30. Substituting these numbers into the appropriate derived equation and requiring that it be 90% probable yields a remarkable result. Such coincidences would happen anywhere from once a week to once every other month. Most people’s fuzzy math would probably have estimated a much longer time period.
If you are curious about how often you might expect such coincidences to occur, e-mail your numbers for N, k, and L to me and I’ll calculate the estimate for your case and send it to you.
Next time you get that “Ralph” call, rather than attributing it to ESP, you might tell Ralph: “Hey, I was just thinking about you, so you can consider yourself my coincidence of the month.”
Featured image: Ancient Planet by PIRO4D. Public domain via Pixabay.