Gambler's Fallacy

The gambler's fallacy is the expectation that an independent random process will correct a recent imbalance, so that after a run of one outcome the other becomes more likely. The classic case is the coin: after five tails, the next flip feels overdue for heads, and it is not, because the coin has no record of the five.

Amos Tversky and Daniel Kahneman gave the standard account of it in "Belief in the law of small numbers," published in Psychological Bulletin in 1971. Describing experiments in which people predict events in a randomly generated sequence, they wrote that "subjects act as if every segment of the random sequence must reflect the true proportion: if the sequence has strayed from the population proportion, a corrective bias in the other direction is expected. This has been called the gambler's fallacy." Note the phrasing of that last sentence: the name already existed, and what Tversky and Kahneman supplied was the mechanism behind it. "The heart of the gambler's fallacy is a misconception of the fairness of the laws of chance," they went on. "The gambler feels that the fairness of the coin entitles him to expect that any deviation in one direction will soon be cancelled by a corresponding deviation in the other. Even the fairest of coins, however, given the limitations of its memory and moral sense, cannot be as fair as the gambler expects it to be."

The underlying error is about sample size, and Tversky and Kahneman named it precisely. Their subject was what they called the law of small numbers, the belief "that the law of large numbers applies to small numbers as well." Large samples really do come to resemble the population they are drawn from; that is a theorem. The intuition wrongly extends the guarantee to short runs, and once a short run is expected to be representative, any deviation in it looks like something that must be undone. The gambler's fallacy is that expectation applied forward in time.

Its mirror image is recency bias, and holding the two together is the most useful thing on this page. Recency bias reads a run as a signal that the run will continue, so a good three years means expect a good fourth. The gambler's fallacy reads the same run as a signal that it will reverse, so a bad three years means expect a good fourth. They are opposite forecasts produced by the same underlying mistake, which is treating a short stretch of a noisy series as though it carried more information than it does. Neither is a reading of the evidence; both are ways of not needing one. A reader who has decided that recency bias is the error to avoid, and who therefore bets against whatever has recently happened, has adopted the gambler's fallacy in its place.

The financial version needs one honest qualification that the coin case does not. A coin's flips are independent by construction, so the fallacy is demonstrably a mistake there. Whether stock returns are independent across periods is a genuine and long-running empirical question, and this page takes no position on it. That is exactly why the fallacy is defined by its reasoning rather than by its conclusion. Someone who buys after a fall because the valuation is now attractive on evidence they can state has made an argument that may be right or wrong. Someone who buys because the fall makes a rise "due" has made no argument at all; they have asserted an entitlement to a correction, which is the thing Tversky and Kahneman identified. The conclusion can be identical and only one of them is a fallacy.

Two shapes it takes with money, both of which involve committing more. The first is averaging down on a losing position on the strength of the loss itself, which increases the size of the position at the same time as the reasoning behind it weakens. The second runs the other way, treating a stretch of good years as debt that will be called in, and reducing risk not because the plan changed but because the good years feel used up. Both convert a feeling about a short sequence into a change in exposure.

Knowing about it does not dissolve it. Tversky and Kahneman were explicit that "related biases, such as the gambler's fallacy, survive considerable contradictory evidence," and their paper is largely about professional researchers making the error while knowing the statistics. What actually interrupts it is a rule decided in advance, because a rule does not have to be re-argued at the moment the run feels longest.

The arithmetic that dissolves the coin version is worth walking through, because the two probabilities people confuse are both correct.

A fair coin has come up tails five times. Two different questions have two different answers.

• What is the chance the next flip is heads? One in two. The coin's mechanism does not change, and the five previous flips are not inputs to it.
• What was the chance, before any of them, that six flips in a row would all be tails? One in 64, since each flip halves the probability and 2 to the sixth power is 64. Equivalently, the chance that six flips contain at least one head is 63 out of 64.

Both statements are true at once. The rare thing is the sequence viewed in advance; the next flip viewed now is an even bet. The gambler's fallacy is the second answer being smuggled into the first, so the unlikeliness of the whole run gets attached to the one flip that has not happened yet.

A hypothetical example of the money version, and of why the arithmetic can be right while the reasoning is wrong. Bea owns 100 shares bought at $50, a $5,000 position, and the price falls to $25. She buys 200 more shares at $25, another $5,000. She now owns 300 shares for $10,000, so her average cost is $10,000 divided by 300, or $33.33 a share.

Every part of that is arithmetically correct, and it does lower the price at which she breaks even, from $50 to $33.33, which is a rise of about 33 percent from $25 rather than the 100 percent she would have needed on the original position. What it also does is double her exposure to a single holding at the moment she has the least evidence that it will recover. If her reason is a view she can state about the business or the valuation, that is an investment decision with an argument behind it. If her reason is that the stock is "due," the arithmetic has not supplied the argument; it has only made the position larger.

Why the concept earns its place

• It names a specific, testable reasoning error rather than a vague failing, and the test is whether a correction is being treated as owed or as evidenced.
• It has an unusually clean intellectual history. Tversky and Kahneman's account is short, readable and still the standard one.
• Pairing it with recency bias covers both directions in which a short run gets over-read, which is more useful than either concept alone.

The traps in applying it

• Financial returns are not coin flips, so labeling someone's contrarian decision a fallacy requires knowing their reasoning, not just their trade.
• The countermeasure is easy to overcorrect into. Deciding that runs always continue is recency bias, and it is the same misreading pointed the other way.
• Averaging down is arithmetically sound as a way to lower an average cost and simultaneously increases concentration, so the calculation cannot settle whether it is a good idea.
• Recognizing the bias does not remove it. Tversky and Kahneman noted that biases of this kind survive considerable contradictory evidence, including among people trained in statistics.