Thunder Bay, Canada. Used with permission of Dan Hunt, MD.
Source: Thunder Bay, Canada. Used with permission of Dan Hunt, MD.

Coincidences attract our attention because they seem weird, odd, or unlikely. Their improbability stimulates wonder—“what are the odds of this happening?” Probability theory, which is a branch of mathematics, promises to shed light on this defining aspect of meaningful coincidences. 

Carrying this banner, I valiantly strode forward into probability’s thicket of mathematical concepts and formulas and emerged to write a report. My confidence was bolstered by having gotten a Double A in a probability theory seminar at Swarthmore College. 

Here’s what I found in the thicket.

Why have we developed probability theory?

The purpose of probability theory is to understand randomness and uncertainty to enable us to more accurately predict the future. 

Probability is the measure of the likeliness that an event will occur. It is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain we are that the event will occur. 

A simple example is the toss of a coin. The probability is 1/2 (or a 50% chance) of either "heads" or "tails". 

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. Generally speaking, statistics is concerned with the past (the collected data) while probability is concerned with the future (the likelihood that something will happen). Probability usually, but not always, relies on data from statistics to make its predictions. 

Probability theory began in the 16th century in attempts to analyze games of chance like dice, playing cards, and roulette wheels.. Since then several sometimes conflicting methods of assessing probability have been developed. The two most prominent current methods  are the “frequentist” and Bayesian methods. The frequentists rely on the familiar bell shaped curve while Bayesians insist that subjective estimates of probability play an important role in determining a final probability. While much of the math we are taught has the ring of eternal certainty, in probability theory there remains some conflict.  

Figuring out the probability of coincidences

Given the importance of probability in defining coincidences, I combed some of the simpler math literature to come up with an approach. I concluded that base rates, personal meaning and time windows contribute to the probability of a coincidence.  

Base rates. The probability of a coincidence is calculated by first looking at the base rate (frequency of occurrence) of each of the two independent events. For example, you think of a friend whom you have not thought about in years, and that friend contacts you shortly afterwards. What are the odds of these two events coinciding?

The base rate of a specific thought is the number of times that thought occurs divided by the total number of thoughts. So this single thought about your friend, measured against all the thoughts about other people you’ve had over the years, has a pretty low base rate. Let’s arbitrarily estimate a low base rate for this thought at 0.005, or 5 out of 1000 thoughts about people. 

We could similarly estimate the base rate of contact from that friend at 0.01, or 1 time out of 100 contacts. Then the probability of the coincidence is 0.005 x 0.01, or 0.0005 which is quite low, but certainly possible.

Time windows.  Then consider the window of time between the thought of your friend and the friend contacting you. A small time window (a few minutes or hours) makes the coincidence less probable than an interval of a few days or weeks. We also factor in the length of time since you last communicated with your friend.  A week since the last contact would make the coincidence far more likely than several years since the last contact. 

Personal meaning. And what about the importance of the contact to you and your friend? If your friend communicates something of vital importance to you,  the coincidence becomes more improbable than if the friend just wanted to say “Hi”?

Even if we could solve this problem with complex equations involving time windows and personal meaning, a more fundamental problem precedes the development of the complex equations: how do we determine the actual base rates of each leg of the coincidence? What is the base rate for thinking of a friend? What is the base rate for contacts by friends from whom you have not heard for a “long” time?

The calculation of the probability of this common coincidence becomes a daunting task. 

Also, different kinds of coincidences require different probability approaches. For example, “meeting someone who helps you to advance in work, career or education” will require different formulas than coincidences involving people who have numerous characteristics in common, like doppelgangers. 

In future posts I will explore how statisticians, psychologists and pragmaticians look at coincidence probabilities.  

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.

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