Can You Accurately Estimate Coincidence Probabilities?

Statisticians who study coincidences are fond of saying that ordinary people don’t know how to estimate the probabilities of coincidences. To illustrate our poor statistical thinking, statisticians trot out the Birthday Problem: How many people would have to be in a room to have a 50 percent probability that any two of them have the same birthday?

Non-statisticians do not confront problems like this in everyday life or when judging the probability of a coincidence. It’s a brainteaser that requires sophisticated mathematical reasoning.

The answer is a much smaller number than your intuition might guess: 23. Remember this is not about a 100 percent probability. It is a 50 percent probability. With two rooms of 23 people, on average one room would contain a match. This is a tough problem to solve!

Tasks like the birthday problem have been used to support the claim that we misjudge the likelihood of coincidences. Because we don’t have a good grasp of the likelihood of events, the reasoning goes, we overestimate the rarity of coincidences and treat them as more meaningful than they actually are.

Many researchers (Hanley, 1984, 1992; Mathews & Stones, 1989; Mock & Weisberg, 1991) have reported that people’s experiences of coincidences tend to reflect an underlying problem with their probabilistic reasoning. Consequently, much has been made of the association between people’s ability to solve classic decision making tasks and their reported frequency of coincidental experiences (Blagrove, French & Jones, 2006; Blackmore, 1997; Blackmore & Trosiancko, 1985; Brugger, Landis & Regard, 1990; Bressan, 2002; Dagnall, Parker, & Munley, 2007; Musch & Ehrenberg, 2002). As it stands, despite the wealth of studies examining this relationship, it is not clear whether the frequency of experienced coincidences is associated with poor probabilistic reasoning or a failure in understanding random sequences.

Susan Jane Blackmore at the University of Plymouth and her colleagues have shown that people who tend to hold strong beliefs in the paranormal also tend not to be good at tests of probabilistic reasoning, or generating and spotting randomness in series of numbers. And a 2014 study by Robert Brotherton at Goldsmiths University of London and Christopher French at Goldsmiths University of London shows that people who hold strong beliefs in conspiracy theories tend to make more errors in understanding statistical concepts.

Magda Osman and colleagues (Mark Johansen Cardiff University, and Christos Bechlivandis University College London) have been working on several empirical investigations looking at people's coincidental experiences. In one of the projects, the task involved asking people to record their coincidences for periods of five weeks. The study did not define coincidences, instead it was left up to participants to decide what they considered coincidences to be for themselves. The idea was to look at coincidences in the wild rather than create fictitious coincidences to study, such as the birthday problem.

The researchers compiled a set of real world coincidences reported in the diary study. They then asked a different set of participants to rate them by the probability of the coincidences occurring, and how likely they were to occur by chance. The participants were remarkably consistent in their ratings. For each type of judgment, regardless of the fact that people differed in age, gender, educational background, they gave similar judgments on how coincidental the set of difference coincidences were. They also had similar judgements as to how likely they thought the coincidences were to occur, as well as their ratings of possible causality.

Why is showing high levels of convergence in different judgements about likelihood and causality noteworthy? Without telling people what coincidences are, or how to interpret them, or giving any benchmark of rarity, this work shows that people are fundamentally attuned to judging the likelihood of various patterns of recurring events in similar ways. This challenges conventional academic wisdom that poor probabilistic reasoning leads people to misjudge the probability of coincidences. People do vary in the kinds of coincidental experiences they have, and the frequency by which they have them. However people tend to agree on what makes a coincidence highly unlikely or not. This requires some basic sense of probabilities in the world.

When looking at other people's coincidences, humans are good probability estimators. When estimating the probability of their own coincidences, the story changes.

The researchers couldn't do a proper comparison between personal judgments and others judgments of the same coincidences. The judgement questions were changed during the main experiments compared to the initial diary questions. In later studies they found a classic egocentrism bias suggesting that people inflate the rarity of their own personal experiences of coincidences compared to others judging the same coincidences.

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Take home message: To accurately judge the probability of your own coincidence, examine it as if you were another person separated from the emotional charge.

This post was co-written with Dr. Magda Osman, an associate professor at Queen Mary University of London.