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  • Writer's pictureBernard Beitman, MD

A Statistician’s Approach to Coincidences: What Are the Odds? (Part 1)

Updated: May 30, 2023

Clarifying the probability of coincidences helps to better understand them.


Co-authored by Tara MacIsaac, a reporter for the Epoch Times and editor for the Beyond Science section. She explores the new frontiers of science, delving into ideas that could help uncover the mysteries of our wondrous world.


Unlikeliness characterizes coincidences. A common kind of coincidence, for example, is one in which you think of a friend and that friend calls you. Your first thought might be, “What are the chances?”

In the previous post, we bumped into difficulties estimating the probability of this coincidence.

The main issue is that there are so many unique variables for each situation; it’s difficult to estimate the rate of occurrence (base rate) for each part of the coincidence. How long has it been since the friend has contacted you? How often do you think of the friend? Many more intricacies complicate the issue.

Estimating the probability of other coincidence types seems equally, if not more, difficult. Since unlikeliness characterizes coincidences, clarifying their probabilities is a necessary task in better understanding them.

If it is so difficult to calculate coincidence probabilities what then? There seem to be at least three ways out: the statistical, the psychological and the practical. Each makes a contribution to the promise and problems of probability. In this post, I start with those who should know—statisticians.

Statisticians who study coincidences generally believe that “ordinary” people do not know how to judge probability. Statisticians often use the birthday problem to illustrate their point: “How many people need to be in a room to have a 50% probability that any two of them will have the same birthday?” Most people guess numbers that are much too high. The answer is 23.

The first common mistake made by “ordinary” people is to misunderstand the question. We think the question is: “How many people need to be in a room for two of them to have the same birthday, like my birthday.” We assume that the birthday to be matched has already been selected.

With this assumption, over 100 is a pretty good guess. Why? Because specifying the birthday, makes the probability much lower. Not specifying the birthday means that any birthday will do. That increases the probability.

So our first problem is that we don’t hear the question correctly.

A second common mistake is to ignore the 50% requirement. The form of the answer is unfamiliar to most of us: out of 100 rooms with 23 people in each, only ½ will have two people with the same birthday. We are not used to thinking of answers to probability questions like this.

Third, while there are several ways to solve this problem, the easiest way is to assume that there is no match and begin the calculations from this assumption. Not many of us would think of solving the problem this way.

Using the birthday problem, statisticians conclude that we don’t understand probability based on a kind of question most of us have not attempted to solve.

The birthday problem doesn’t prove that people over-estimate the improbability of their coincidences. But statisticians have a better argument when it comes to our tendency to neglect the base rate. When we neglect the base rate we become focused on the unlikeliness of the current event and do not appreciate the frequency of events like it.

I will explore this more in my next post.

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