The gambler's fallacy: streaks, variance and independent bets
Why a streak can't change the next bet, the 1913 Monte Carlo story, the hot hand, and worked odds for losing streaks and short sessions.
Fact-checked 18 October 2026 · 5 sourcesAfter a long run of one result, the other one starts to feel due. It isn't. When each bet is independent, the chance on the next one is exactly what it was before the streak began, and this article works through the numbers to show why, and what streaks and short-term luck really look like next to the house edge.
What the fallacy is
The gambler's fallacy is the belief that an outcome which has come up less often than expected is now more likely, or that one which keeps coming up must soon stop. It is also called the Monte Carlo fallacy or the "maturity of the chances".
The mistake is about independence. Two results are independent when one tells you nothing about the other. A roulette wheel, a fair die or a well-built random number generator has no memory: whatever happened before, the next result is drawn from the same odds.
Monte Carlo, 1913
The story that gave the fallacy its nickname is set at the Casino de Monte-Carlo on 18 August 1913. Black came up 26 times in a row, and as the run grew, players piled onto red and lost millions of francs. The account most writers use comes from Darrell Huff and Irving Geis's 1959 book How to Take a Chance, which describes "a near-panicky rush to bet on red" from about the fifteenth black. The book appeared 46 years after the date it describes, so treat the details as a well-told story rather than a record; the arithmetic it illustrates holds either way.
A single-zero wheel has 37 pockets, 18 of them black, so each spin lands on black with probability 18/37, or 48.65%. Here is how rare the run was, and how little that matters:
| Question | Probability | |---|--:| | 26 blacks in a row, judged before the first spin | (18/37)^26 ≈ 1 in 136.8 million | | 26 in a row of either colour | 2 × (18/37)^26 ≈ 1 in 68.4 million | | Black on the next spin, after 25 blacks | 18/37 = 48.65% | | Black on the next spin, after 25 reds | 18/37 = 48.65% |
Huff and Geis wrote that after the twentieth black, players believed there was "not a chance in a million of another repeat". The real chance was 48.65%, the same as on the first spin. The run was rare as a whole sequence, but once 25 spins have happened, only one spin is left to decide.
The hot hand: the opposite mistake
The mirror image of the gambler's fallacy is the hot hand: believing a streak will continue because you're on a roll. In skill-based activities this isn't always wrong. A famous 1985 study by Gilovich, Vallone and Tversky concluded that basketball fans' belief in shooting streaks was an illusion, but in 2018 the economists Joshua Miller and Adam Sanjurjo showed that the way streaks had been measured contained a subtle bias, and that correcting it reversed the study's conclusion. As the BBC's Tom Stafford noted in a 2015 article, in most of life streaks are linked for good reason: some days your tennis serve really is working.
A random game has nothing like that. For truly random events like roulette or the lottery, Stafford wrote, "there is no force which makes clumps more or less likely to continue."
Real gamblers show both biases anyway. Rachel Croson and James Sundali coded security video of 18 hours of play at one roulette table in a Reno casino: 139 players and 24,131 bets. After six or more of the same outcome in a row, 85% of even-money bets went against the streak. And of the 139 players, 111 left the table right after a losing spin, against 28 after a win.
Streaks are normal
Long streaks feel like evidence that something has changed. Usually they're just what randomness looks like. Take a dice bet at 2× with a 1% house edge, which wins 49.5% of the time and loses 50.5%.
Ten losses in a row starting from a given bet is unlikely: 0.505^10, about 1 in 927. But you don't place one bet, you place hundreds, and somewhere among them a streak is likely to start:
| Losing streak of at least | Within 100 bets | Within 1,000 bets | |---|--:|--:| | 5 | 82.4% | over 99.9% | | 7 | 33.4% | 98.6% | | 10 | 4.8% | 41.3% | | 15 | 0.2% | 1.7% |
These figures are exact, worked out by tracking the length of the current losing run bet by bet, and a 200,000-session simulation of several of them agreed to within 0.1 percentage points. A ten-loss streak turns up somewhere in about four out of ten 1,000-bet sessions. That is also why systems that double the stake after each loss eventually hit a streak they can't afford; our article on the Martingale works through the bet sizes.
The same goes for a 15-slot wheel with 7 T, 7 CT and one bonus slot. Five T results in a row, starting from a given spin, has a 2.2% chance, about 1 in 45, and going 50 spins without the bonus has a 3.2% chance. Neither makes the next spin different: T is still 7 in 15 and the bonus is still 1 in 15.
Variance in the short run, the edge in the long run
Variance is how far results scatter around the average. It is the reason a session can end up, and the house edge is the reason the average is below zero. They grow at different speeds. Here is the same 2× bet, one coin per bet:
| Bets | Average result | Typical swing (one standard deviation) | Chance of being ahead | |--:|--:|--:|--:| | 10 | −0.1 | ±3.2 | 36.5% | | 100 | −1 | ±10 | 42.1% | | 1,000 | −10 | ±31.6 | 36.4% | | 10,000 | −100 | ±100 | 15.6% | | 100,000 | −1,000 | ±316 | 0.1% |
The average loss is 1% of everything wagered, so it grows in step with the number of bets. The swing grows only with the square root: 100 times more bets means 10 times more swing. Over a short session the swing dwarfs the edge, so being ahead is common; over enough bets the edge takes over. (For 10 bets, a further 24.6% of sessions end exactly level.) A good session doesn't mean the odds have turned, and a bad one doesn't mean a win is owed. House edge and RTP explains how the edge is calculated for each game.
Why provably fair bets are independent
On cs2.casino, solo games such as Dice and Plinko compute each result from three inputs: a server seed committed by its SHA-256 hash before you bet, a client seed you can change, and a nonce that goes up by one with every bet. The randomness is HMAC-SHA256(server seed, "client seed:nonce:k"). Your previous results are not an input. The roll after 25 losses is computed exactly like the first one, just with the next nonce. Once you rotate the seed pair, the old server seed is revealed and you can recompute any bet made with it on the provably fair page; provably fair explained covers how the commitment works.
Because each result depends only on those seeds and the nonce, no betting pattern can change it. The useful response to a streak is not to change your bet size but to stick to the limits you set before you started.
