The gambler's fallacy

The fallacy is not stupidity and it is not confined to gamblers. It is a misapplied version of something true: over very many trials, proportions do settle down toward their underlying probability. The error is in the timing and the mechanism. Settling down happens by dilution across thousands of later trials, not by correction on the very next one.

What the gambler's fallacy is

The gambler's fallacy is a probability error that arises when a person treats independent events as if they were connected. Two events are independent when the outcome of one leaves the probability of the other unchanged, which is true of coin flips, roulette spins, dice rolls and lottery draws, and false of card draws from a deck that is not reshuffled.

What distinguishes the gambler's fallacy from a simple miscalculation is that the arithmetic feeding it is usually correct. The probability of six heads in a row really is 1 in 64, or 1.6 percent, because 2 to the power of 6 is 64. Someone who has just seen five heads and concludes that a sixth is unlikely has used a real number. They have attached it to the wrong question, and the difference between those two questions is the whole of this page.

The one-sentence version: the coin has no memory

In one sentence: the probability of a sequence and the probability of the next event given the sequence so far are different numbers, and only the second one is about what happens now.

Recognize the fallacy by its verbs. Phrases such as "due", "owed", "overdue", "has to even out", "cannot keep going" and "law of averages" all assert that past outcomes exert pressure on future ones. Applied to independent trials they are always wrong. Applied to a shrinking deck, a finite raffle drum or a machine that wears as it runs, the same phrases can be exactly right, because the underlying probability genuinely does change. The first job is always to ask whether the events are independent, and the second job is to ask how you know.

The arithmetic worked through: five heads, then a sixth

Work the arithmetic on a fair coin flipped six times, where the first five have all landed heads. Two questions look identical in conversation and have different answers.

• Before any flips: the probability of six heads in a row is one half multiplied by itself six times, which is 1 in 64, or about 1.6 percent.
• After five heads: the probability that the sixth is a head is one half, or 50 percent, because the five heads have already happened and their probability is now 1.
• The two numbers differ because 1 in 64 is the chance of the whole sequence including the part you have already watched, and 1 in 32 of that 1 in 64 was consumed by the first five flips.
• Check it: 1 in 32 for five heads, multiplied by one half for the sixth, gives 1 in 64. The rare part was getting here, and getting here is no longer in question.

Now the dilution. Suppose the first ten flips are all heads, which is 100 percent heads and looks badly out of line. Flip 990 more times. The expected number of heads in those 990 is 495, so the expected total is 505 heads in 1,000 flips, or 50.5 percent. Carry on to 10,010 flips and expect 5,010 heads, or 50.05 percent. At no point did a compensating run of tails arrive. The ten-head surplus stayed exactly where it was, worth ten flips, and became invisible because the denominator grew.

The case usually cited as the historical illustration is a roulette table at the Monte Carlo Casino in August 1913, where black is reported to have come up 26 times in succession while players bet increasingly heavily on red in the belief that it was due. On a European wheel with 37 pockets, 18 of them black, the probability of that specific run is 18 divided by 37, raised to the power of 26, which works out at roughly 1 in 140 million. That figure describes the run before it began. It says nothing about spin 27, which was still 18 in 37 for black, and the money lost that night was lost to a probability that never changed.

What a failure looks like: betting on the correction

A gambler's fallacy failure looks like a decision whose entire justification is the record of previous outcomes. It is not confined to casinos, and the versions outside them are more expensive because they are not recognized as bets.

• Choosing lottery numbers that have not appeared recently, on the reasoning that they are overdue, when every combination is equally likely on every draw.
• Concluding that a run of four quiet quarters means a bad one is coming, where the quarters are governed by conditions that have not been examined at all.
• Assuming that an inspection process which has passed 50 consecutive batches must be about to find a fault, rather than asking whether the inspection still works.
• Treating a run of three successful hires as evidence that the next hire is more likely to fail, on the basis that luck is being used up.

The third example is the interesting one, because there the reasoning points at a real question and gets there by the wrong route. A long run of passes is worth investigating, not because a fault is due, but because a process that never fails may not be measuring anything. That is a hypothesis about the mechanism, and it is tested by examining the mechanism, which is what the null hypothesis is for.

The common misreading: the law of large numbers and the sunk cost

The common misreading is that the law of large numbers supports the fallacy. It does not, and the difference is precise. The law of large numbers says that as the number of independent trials grows, the observed proportion converges on the true probability. It makes no claim about any individual trial and it contains no correcting force. The gambler's fallacy asserts that the next trial is affected. One is a statement about limits, the other is a statement about now, and the arithmetic above shows convergence happening by dilution rather than by compensation.

The gambler's fallacy is also confused with the sunk cost fallacy, which sits nearby at the same table and is a different error. The sunk cost fallacy is continuing an unpromising course because of what has already been spent on it. The gambler's fallacy is continuing because of what the pattern of outcomes seems to promise. One looks backward at investment, the other looks backward at frequency, and a losing streak often produces both at once.

A third misreading runs the fallacy in reverse. The hot hand belief holds that a run of successes makes further successes more likely, which is the opposite prediction from the same evidence. It deserves an honest treatment rather than a dismissal.

The hot hand, and why it is not a settled dismissal

The hot hand question is where this topic stops being tidy. Thomas Gilovich, Robert Vallone and Amos Tversky published an analysis of basketball shooting in 1985 concluding that streaks were a cognitive illusion, and for three decades the hot hand was taught as a companion error to the gambler's fallacy. Then the economists Joshua Miller and Adam Sanjurjo showed, in work published in 2018, that the standard method of measuring streaks contains a subtle selection effect that biases it against finding them. Correcting for that bias, the evidence leaves room for a modest real hot hand in some skilled physical tasks.

The distinction that survives is between mechanisms. A coin, a die and a roulette wheel have no state that a previous outcome could alter, so streaks in them carry no information and the gambler's fallacy is simply wrong. A person shooting a basketball has muscles, attention and confidence, all of which are physical states that could plausibly persist for a few minutes. Whether they do, and by how much, is an empirical question that is still being argued. Saying so is more useful than declaring both beliefs fallacies, and it points at the right first question: is there a mechanism by which the past could reach the present?

The test to run on any streak

Run these four questions the next time a run of outcomes is used to predict the next one.

1. Ask whether anything physical connects one trial to the next, and name it if so.
2. Ask whether the probability of the next event has changed, or only your sense of how the sequence looks.
3. Ask how many sequences were watched before this one seemed remarkable, since somewhere a run of ten is always in progress.
4. Ask what the correction is supposed to be made of, if the claim is that the outcome is due.

The fourth question ends most versions of the argument, because there is no answer to it that does not invent a mechanism. Converting the run back into counts and proportions is the habit taught in Reading numbers, deciding whether a run exceeds what chance produces is the job of Statistical significance, and the assumption a test of the run would be examining is set out in The null hypothesis. The straw man fallacy is worth knowing here too, since the argument against a streak claim is often aimed at a stronger version of it than anyone stated.

Why this phrase also returns a television episode

Searches for this term also surface a crime drama episode that borrowed the name, which is why results mix reasoning material with cast lists and episode guides. That is a naming coincidence and nothing more. This page is about the probability error: what it claims, why it is wrong for independent events, and where a run genuinely does carry information.

The calculator for this is one line, and it returns the same answer every trial

The calculator for the gambler's fallacy is a single multiplication, which is why no site sells one. The probability of a run of k identical outcomes is p multiplied by itself k times, so six heads is 0.5 to the sixth power, or 1 in 64, and twenty-six blacks on a European wheel is 18 over 37 raised to the twenty-sixth power, or roughly 1 in 140 million. The probability of the next trial is p. It was p before the run, it is p during the run, and it will be p after it. That constant is the entire content of the topic and a calculator cannot make it more interesting.

One decision has to be made before the formula applies, and no tool can make it for you: whether the trials are independent. The multiplication assumes independence, accepts your numbers without checking, and returns a confident answer whether the assumption holds or not. Cards dealt from an unshuffled deck, tickets drawn from a finite raffle drum and a machine that wears as it runs all violate it, and for those the probability genuinely does change from trial to trial. Deciding which case you are in is a question about the mechanism, and the mechanism is not one of the inputs.

A second decision matters more and is easier to miss: whether the run was specified before it happened or noticed afterward. Those are different questions with different answers. The probability of six heads on the next six flips is 1 in 64. The expected number of runs of six or more heads somewhere in 100 flips is 1 over 64 for the opening position plus 94 chances at 1 over 128 each, which is 0.75 expected runs, so a run of six turns up in roughly half of all 100-flip sequences. The first number describes a prediction. The second describes a discovery, and a run discovered after the fact is nearly always the second kind.

What the calculation does not settle is whether the run means anything, and that is the whole of the disagreement it gets used in. A probability tells you how often chance produces a pattern; it does not tell you whether chance produced this one. Answering that needs a mechanism by which the past could reach the present, and where such a mechanism plausibly exists, as in the hot hand question, the arithmetic alone cannot close the case in either direction.

Where to go next

NUMBERSReading numbersThe checks to run on any statisticNUMBERSThe null hypothesisWhat a test is actually testingNUMBERSStatistical significanceStatistical significance and what a p value really meansFALLACIESThe straw man fallacyDefinition and examples