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1Fix the driver behind crashes, sound loss and screen glitches2Repair Windows errors before they cause bigger problems3Scan for outdated or missing drivers - takes under a minuteA rare event can happen without being impossible—and a coincidence that looks astonishing after the fact may not have been unlikely in the broader set of things that could have happened. The key is to define the event before judging its odds, then account for how many opportunities and possible matches were in play.
What does “impossible” mean in probability?
Once an outcome has happened, it can feel as though that exact result was destined to be extraordinarily unlikely. But if a process has a complete set of possible outcomes, one of them has to occur. The fact that the realized result was only one possibility does not, by itself, make it inexplicable.
David J. Hand’s book The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day describes statistical principles that help explain why rare events and coincidences arise. The book’s publisher summarizes five central laws: inevitability, truly large numbers, selection, the probability lever, and near enough. Kevin Gray and Cannon Gray’s 2017 KDnuggets article “Stuff Happens: A Statistical Guide to the ‘Impossible’” discusses those ideas alongside additional statistical cautions.
Why rare events happen somewhere
Inevitability: something in the outcome space must occur
Consider a lottery draw with a fixed set of possible number combinations. Any single combination may be very unlikely before the draw, but one combination will be drawn. Afterward, calling the winning combination “impossible” confuses the probability of that exact, preselected combination with the certainty that some combination would win.
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Truly large numbers: opportunities accumulate
A small chance on one try can add up across many tries, people, places, or time periods. The relevant question is not only “How unlikely is this on one opportunity?” but also “How many opportunities were there for it to happen?”
Imperial College London quotes Professor David Hand: “The law of truly large numbers says that even an outcome that has a tiny chance of occurring can become almost certain if you give it enough opportunities”. Hand’s explanation at Imperial College London captures why a rare event can be expected to occur somewhere across a sufficiently large set of opportunities.
Why a coincidence looks more surprising after it happens
Selection: the match may be chosen after the search
There is an important difference between predicting a particular event in advance and noticing a striking match after looking across many events. If someone names one exact result before a lottery draw, the probability concerns that particular result. If, after the draw, they search many past draws, number patterns, dates, people, and descriptions for something that fits, the probability of finding some attention-grabbing match is a different question.
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This is selection: the highlighted coincidence may be only one of many comparisons that could have been made. The more possibilities someone searches, the less meaningful it is to assess the eventual match as though it had been specified in advance.
Near enough: relaxed criteria create more matches
A coincidence can also look exact because its criteria were loosened after the fact. A prediction that counts as a match only when every detail is correct is narrower than one that accepts an approximate date, a similar description, or a partly correct result. Define what counts as a match before calculating its probability; otherwise “close enough” can quietly expand the set of outcomes that qualify.
Five questions to test an “impossible” event
Before treating an unusual occurrence as evidence of something extraordinary, work through the assumptions behind the claim:
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- Was the event defined in advance? A precise prediction made beforehand is not equivalent to a pattern selected after it occurred.
- How many opportunities were available? Count the relevant people, trials, locations, time periods, and outcomes—not only the one case that drew attention.
- How many possible matches were considered? A result found after searching many descriptions or comparisons has a different chance of appearing than one chosen beforehand.
- Are the trials independent, and is the model appropriate? Dependence between events and the assumed probability distribution can change the calculation substantially.
- Were the match rules kept fixed? Check whether a near match, partial hit, or revised description was accepted only after the result was known.
Why probability assumptions can change the answer
The probability lever: the model matters
A probability is not meaningful without assumptions about the outcome space and the process that generates results. Independence, distribution, and event definition are not technical footnotes: changing them can change the estimated odds. A calculation based on one model should not be presented as a universal probability for every setting.
Gray and Gray’s 2017 article illustrates this with a comparison about a “5-sigma” event. Under the normal-distribution setup described in that article, it gives an estimate of about 1 in 3.5 million; under the contrasting Cauchy-distribution setup, it gives about 1 in 16. Those figures illustrate how a selected distribution affects a calculation. They are not universal estimates of financial-crash risk or official observed rates.
Independence cannot simply be assumed
If several events are treated as independent when they are connected, multiplying their individual probabilities can produce a misleading result. The Significance interview with Hand discusses unjustified assumptions of independence in coincidence reasoning. The interview is a useful reminder to ask whether the events could influence one another or share a common cause before combining their odds.
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What the familiar examples do—and do not—show
Paul the Octopus and a run of correct predictions
Gray and Gray’s 2017 article gives a probability of 1/256 for Paul the Octopus to predict all eight of the cited World Cup matches correctly under the article’s setup. That is an illustrative calculation for the specified sequence and assumptions, not a verified organizational statistic or a general rate for prediction accuracy. To judge how surprising the story is, one would also need to consider how the predictions were defined, what counted as a correct selection, and how many other predictions or animals might have been noticed.
Multiple lightning strikes and repeated lottery wins
Stories about someone being struck by lightning more than once or winning a lottery more than once naturally focus on the person who experienced the event. That is a selected case. To assess its likelihood, the calculation would need to account for the number of people exposed, the number of relevant opportunities, the event’s definition, and whether repeated exposures are independent. A striking anecdote alone does not supply those inputs.
Small samples, data dredging, and regression to the mean
An eye-catching pattern in a small sample may be unstable: a few observations can look meaningful even when they do not establish a general rule. Searching many variables, groups, time windows, or outcomes for a notable association—often called data dredging—raises the chance of finding something that looks significant by chance. If a pattern was discovered through a broad search, it needs an appropriate analysis or replication before it can support a broader conclusion.
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Regression to the mean is another reason not to overread an extreme result. When an observation is unusually high or low partly because of chance variation, a later observation will often be closer to the average even without a special intervention. That shift does not, on its own, establish that a hidden force caused the initial extreme result or its later change.
How to interpret a coincidence without overclaiming
A coincidence can be real and still have a plausible statistical explanation. But probability reasoning alone does not prove that every unusual event has a simple explanation, nor does a coincidence by itself establish supernatural causation, fraud, or a hidden mechanism. The sound conclusion depends on the details: what was predicted, what was selected afterward, how many chances there were, and whether the model fits the process.
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