Gambler's fallacy

Black is due

At a roulette table, the ball has landed on red six spins in a row. A player moves all his chips to black: “It can’t keep doing that. Black’s due.”

The wheel doesn’t know what it did before. The chance of black on the next spin is exactly what it was on the first spin (a little under half, because of the green zero). Six reds in a row looked unlikely before it started, but once it has happened, it says nothing about spin seven.

The number that just came up

A lottery player has played the same numbers for a year. When one of them is drawn this week, she swaps it for a different number: “That one won’t come up again for a while.”

Lottery draws are independent, so a number drawn this week is exactly as likely next week as any other. This isn’t just a table-game habit. In Maryland’s daily numbers game, the amount bet on a number fell sharply right after it was drawn and took months to recover (see Evidence).

The sample that should correct itself

A researcher is testing 50 children chosen at random from a city where the average IQ score is 100. The first child tested scores 150. She expects the average for all 50 to come out at about 100.

Nothing about the remaining 49 children changes because the first scored high; they are still expected to average 100. So the best estimate for all 50 is (150 + 49 × 100) ÷ 50, or 101. Expecting 100 means expecting the rest of the sample to score low to make up for the first child. Tversky and Kahneman used this question in 1971 and reported that “a surprisingly large number of people” gave 100. Nothing is being bet on, and the error still appears.

Four approvals in a row

A loan officer has approved the last four applications in her pile. The fifth looks about as strong as the others, but she finds herself searching harder for a reason to turn it down: “I can’t be approving everything.”

If applications arrive in random order, the last four decisions say nothing about the quality of the fifth. Expecting a run of approvals to be followed by a rejection treats good applications as if they were rationed. This is the least obvious form, because it looks like caution. Researchers have found decisions that swing back after a run in the same direction among asylum judges, loan officers and baseball umpires, a pattern they judged most consistent with the gambler’s fallacy (see Evidence).

• Betting against a streak: the classic form, at roulette tables, in lotteries and in any game with independent rounds.
• Swinging back in judgments: decisions made one after another lean against the previous ones, as in the loan officer example.
• The retrospective gambler’s fallacy: reasoning backwards from a rare-looking outcome. People asked to imagine walking into a casino and seeing a man roll three sixes estimated he’d been rolling longer than people who imagined seeing a less remarkable roll, as if a rare result must come at the end of a long run.
• Expecting small samples to mirror the whole: the sample example above. Tversky and Kahneman called the underlying belief the “law of small numbers”.

• When earlier outcomes use up the supply. Cards dealt from a deck, names drawn from a hat and tickets drawn without replacement really do change what’s left. After many high cards, low cards are more likely.
• When something is actively balancing the outcomes. A rota that gives everyone a turn before repeating, or a manager required to approve about half of requests, makes a run in one direction predict a change.
• When the question is about the long run. Expecting the share of reds to approach its true rate over thousands of spins is correct. What’s wrong is expecting the next few spins to pay back a streak.
• When an extreme result mixes skill and luck. Expecting an unusually good result to be followed by a more ordinary one is expecting regression to the mean, which is sound. It becomes the gambler’s fallacy only if you expect a below-average result to even things out.

The test: does anything connect the earlier outcomes to the next one, such as a finite supply, a balancing rule or a cause that carries over, or am I expecting chance to keep score?

Counting cards

In a card game dealt from a single deck without reshuffling, most of the high cards have already been played. A player bets that the cards still to come will run low.

It sounds like “low cards are due”, but here they really are more likely. Every high card dealt is one fewer left in the deck, so the remaining cards are richer in low ones. The earlier outcomes changed the odds because they used up the supply.

A great first week

A pub quiz team that usually scores around 60 out of 100 scores 85 in the first week of a new season. The captain predicts next week’s score will be closer to 60.

The captain expects the next result to be less extreme, which can look like expecting a correction. But an 85 probably included some lucky questions, and luck doesn’t carry over, so the best guess for next week sits between 85 and the team’s usual level. The captain isn’t predicting a 35 to balance the books. That’s regression to the mean, the case Regression fallacy describes as good reasoning when no cause is invented for it.

Tversky and Kahneman’s account is the law of small numbers: people expect even a short run of a random process to look like the process as a whole, so any few flips of a fair coin “should” be about half heads. A streak looks unrepresentative, and the next outcome feels needed to restore the balance. It’s the same judgment by resemblance, or representativeness, that they used to explain the Conjunction fallacy and Base rate neglect. As they put it, 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.”

Matthew Rabin modeled this as reasoning as if outcomes were drawn from a small urn without replacement: each red taken out leaves fewer reds for later. That’s exactly right for a deck of cards, which is part of why it feels right for a roulette wheel.

Other researchers argue that experience shapes the expectation. Ayton and Fischer propose that it comes from real-world sequences of natural events that do tend to reverse, much like sampling without replacement. Hahn and Warren argue that for someone who experiences random events as a limited stream, with limited memory, some patterns labeled as errors reflect real statistical features of the sequences people actually encounter.

The gambler’s fallacy has a mirror image, the Hot hand fallacy: expecting a streak to continue rather than end. Ayton and Fischer describe the two as opposite expectations about the same kind of sequence. They aren’t necessarily separate mistakes, though. Some researchers argue the hot hand belief grows out of this one: someone who expects random sequences to alternate sees too many streaks in real random data, and explains them as “hot” spells.

• Ask what links the outcomes. If nothing does (a fair wheel, a lottery machine, randomly ordered cases), the last result is irrelevant to the next.
• Think in dilution, not correction. After six reds, the expected number of reds in the next 100 spins is the same as it always was. The streak fades into the total instead of being paid back.
• For decisions made in sequence, judge each case on its own evidence. In the field studies below, the swing-back pattern was weaker among more experienced decision makers, and among loan officers given stronger pay-for-accuracy incentives. Those are observed associations, not tested remedies.

Status: replicates robustly. The gambler’s fallacy is one of the oldest documented biases, and has been found in laboratory tasks, in a representative national survey, in incentivized experiments and in several independent field settings with real money or real stakes. A multi-lab replication with a preregistered protocol reproduced a variant of it at close to its original size. The caveats below concern its explanation and how many people show it, not whether it exists.

Foundations.

• Tversky and Kahneman (1971) described the gambler’s fallacy, citing earlier studies of people producing and predicting random sequences, as an expression of a belief in the “law of small numbers”, the idea that small samples should closely resemble the population they come from.
• Rabin (2002) reviewed the laboratory and field literature, formalized the belief, and noted weaknesses in some laboratory evidence: many tasks weren’t incentivized, and with a fair coin either guess is equally likely to be right.

Surveys and experiments. In a review, Benjamin (2019) summarizes studies that address those weaknesses. Dohmen and colleagues (2009) asked a representative sample of the German population the probability of heads after a specific sequence of eight coin flips: 60% correctly said 50%, 21% gave less than 50% (the gambler’s fallacy direction) and 9% gave more. In incentivized experiments by Benjamin, Moore and Rabin, participants on average gave a 44–50% chance of heads on a first flip but only 32–37% after nine heads in a row. As in Dohmen’s survey, most individual answers were correct, and the wrong answers leaned strongly toward the fallacy.

Multi-lab replication. Oppenheimer and Monin (2009) reported a retrospective gambler’s fallacy: people imagined a longer history of rolling for a man they saw roll three sixes than for one who rolled two sixes and a three. Many Labs 1 (Klein and colleagues, 2014) reran it across 36 samples with about 5,900 participants. The effect replicated in 83% of the samples individually, with a pooled effect (d ≈ 0.61) close to the original (0.69). This tests a close relative of the gambler’s fallacy, a backward inference from the same belief, not the forward prediction itself.

Field evidence.

• Clotfelter and Cook (1993) found that in Maryland’s daily numbers game, money bet on a number fell sharply immediately after it was drawn and recovered gradually over several months.
• Suetens, Galbo-Jørgensen and Tyran (2016), as summarized in Benjamin’s review, used records from the Danish national lottery and found players placed roughly 2% fewer bets on numbers that had won the previous week. The same data showed the opposite pattern after a streak: the longer a number’s run of wins, the more players bet on it (see Hot hand fallacy).
• Chen, Moskowitz and Shue (2016) found decisions negatively related to the previous decision, independent of the merits of the cases, in U.S. asylum courts, in a field experiment with loan officers in India, and in Major League Baseball umpires’ calls on pitches whose location was precisely measured. Umpires were about 1.5 percentage points less likely to call a strike after calling the previous pitch a strike. The effects were stronger after longer runs and weaker with experience and, for loan officers, with stronger incentives. The authors judged the gambler’s fallacy the most consistent explanation but could not fully rule out sequential contrast effects, in which the previous case changes how the current one is perceived rather than what’s expected of it.

What’s disputed.

• The explanation. Ayton and Fischer’s experience-based account and Hahn and Warren’s argument from limited memory compete with the law-of-small-numbers account. Benjamin notes that the Rabin-style models don’t explain everything: in Benjamin, Moore and Rabin’s experiments, people also expected “reversals” among flips taken from scattered, non-consecutive positions, which no internally consistent model of the belief predicts.
• A statistical subtlety. Miller and Sanjurjo (2018) proved that in any finite sequence of fair coin flips, the proportion of heads among the flips that immediately follow a heads is expected to be less than half. So a person who has watched a limited stream of flips may have seen, in their own experience, something like the pattern the gambler’s fallacy expects. Benjamin points out that this doesn’t apply to questions about the chance of heads after a specific sequence, where the correct answer is always 50%.

What remains uncertain is which mechanism produces the belief, and how much it costs in everyday decisions outside gambling and the settings studied so far.