Four bets have lost in a row. The fifth one feels overdue, and the standard correction is that the fifth bet has no idea the other four happened.
That correction is right, and it’s incomplete in a way that matters. Something in the arithmetic of finite sequences keeps handing the gambler’s fallacy evidence, and until 2018 nobody had worked out what it was.
What the Gambler’s Fallacy Claims
The gambler’s fallacy is the belief that a run in one direction makes the opposite outcome more likely next time. Amos Tversky and Daniel Kahneman traced it to the representativeness heuristic in Science in 1974, meaning the habit of expecting a short stretch of results to look like the population it came from. Three years earlier, in Psychological Bulletin, they had given the underlying tendency its name: belief in the law of small numbers. Five heads doesn’t look like a fair coin, so tails feels owed.
The coin has no memory and no ledger. Its next flip is 50/50 the way every flip before it was. The same holds for a point spread, set for one game against one opponent on one afternoon.
Six heads running is unlikely at the outset. Work it out as 0.5 to the sixth power and it comes to 1.56%. After five heads have already landed, that number has stopped being the question. The five are settled and only the sixth is uncertain, so the odds on it are the odds on any flip.
Twenty-Six Blacks in a Row
The other name for the fallacy comes from a roulette table at the Monte Carlo Casino. On 18 August 1913 the ball fell in black twenty-six consecutive times. Run the numbers on a single-zero wheel and a run that long, in either colour, comes to 2 times (18/37) to the twenty-sixth. Call it one in 68 million.
The story goes that gamblers lost millions of francs backing red, betting heavier each spin on the conviction that black had to break. Whether the losses ran to millions is the part nobody has ever documented. It doesn’t need to be true for the anecdote to earn its name. Every spin after the first was priced identically to the first.
Money That Moves on the Fallacy
Charles Clotfelter and Philip Cook put a figure on the behaviour using Maryland’s daily numbers game, publishing in Management Science in 1993. The amount wagered on a given number falls sharply straight after that number is drawn, then climbs back to its old level across the following months.
Nobody in that data is confused about the mechanism. A daily numbers game is transparently random and the players know it. They avoid the recent winner anyway.
Which game it is decides who pays for the belief. The Maryland game pays fixed odds, so avoiding the recent winner there costs a player nothing against any other number. Plenty of lottery games are pari-mutuel, though, splitting a pool among whoever picked correctly. In a pari-mutuel pool, a number everybody has decided is exhausted pays more when it lands, because fewer tickets share the pot. So in those games the fallacy costs the people holding it and quietly funds anyone who ignores it.
Why Experience Keeps Confirming It
Joshua Miller and Adam Sanjurjo found the mechanism and published it in Econometrica in 2018. Flip a fair coin three times. Write down what came immediately after each heads. Repeat that experiment forever and average the results, and the proportion of heads you record settles at 5/12, or 41.7%, against a coin that is honestly 50/50 throughout.
I ran the eight possible three-flip sequences to check, and the fraction is exact. Push the sequence to four flips and the recorded proportion drops to 17/42, about 40.5%, which is the deepest the bias goes. Ask instead what follows a run of three heads inside ten flips and the recorded figure falls to roughly 35%.
Miller and Sanjurjo call this streak selection bias. It exists because choosing which flips to examine on the basis of what preceded them is not a neutral act in a finite record. The magnitude generally shrinks as sequences lengthen and grows as the streak you condition on gets longer.
So the reversal intuition is wrong about the next flip in isolation and right about what turns up when somebody goes back through a finite record and counts. Those two things being true at once is why the belief survives contact with experience. A bettor watching their own results is looking at exactly the kind of short, self-selected record where reversals really do show up more than half the time.
What That Did to the Hot Hand
The mirror-image error is the hot hand, the belief that a run will continue. Thomas Gilovich, Robert Vallone and Amos Tversky examined it in basketball shooting in Cognitive Psychology in 1985. Fans and players believed in it firmly. Shooting percentages, the authors reported, looked independent of previous makes and misses. For three decades it stood as the definitive result.
Miller and Sanjurjo showed the canonical study used the same biased selection procedure their coin example exposes. Corrected for the bias, its long-standing conclusions reverse. The hot hand in basketball shooting now has evidence behind it, and the paper that spent thirty years disproving it was measuring the artefact.
None of which converts into a betting position. A shooting hand is one player under repeatable conditions. A team covering five straight spreads is a different object, priced fresh each week by people who watched the same five games. Our page on line movement covers how little that repricing tells anyone from outside.
What a Covering Streak Is Worth
Team streaks are where this lands for anyone reading a betting board. Seven straight covers looks like a debt coming due. A run that long inside a single seven-game window has a probability of 0.78%, if the team is a coin flip against the number.
Widen the window and the impression falls apart. Give a coin-flip team a full seventeen-game season and the chance it strings together seven covers somewhere in there is 4.67%. Run that across 32 teams and roughly one and a half of them turn up with the streak every year, purely from chance. Runs of five appear in about seven teams a season on the same assumption.
So a covering streak is close to guaranteed to exist somewhere on the board, and the team holding it is the one everybody notices. Whether that team is genuinely mispriced or merely the one the coin favoured this year is a question the streak can’t answer. Our page on why underdogs feel wrong covers how badly people want it to.
Regression to the Mean Is a Different Claim
Regression to the mean gets folded into the fallacy constantly, and it’s a separate and genuinely true thing.
A team that starts 14-2 probably won’t repeat it, and a hitter batting .400 in April probably finishes nearer .280. That happens because extreme results carry luck and luck doesn’t repeat, so later performance sits closer to the underlying ability that was there the whole time. Ability held steady the whole time. What improved was the measurement of it.
Keep the two claims apart. The gambler’s fallacy claims the next flip is likelier to be tails. Regression claims a .400 average was partly noise and the noise won’t recur. One of those claims is false and the other is true, and the true one is already inside the price by the time a line goes up. Betting a team to regress means betting on something the market has read the same way you have, which is what closing line value is for measuring.
Why None of It Helps at the Window
Take the practical question. A bettor winning 55% of the time, which would be an exceptional rate, hits six consecutive losses with probability 0.45 to the sixth, or once in 120 six-bet stretches. Across a 300-bet season the chance of running into at least one such stretch somewhere is 75.2%. Three seasons in four contain one. It’s the ordinary texture of a winning record.
What the fallacy does with that is escalate. Doubling after each loss to recover, at -110 pricing, the sixth bet in the sequence is 42.79 units and the running total staked is 80.68. Reach ten losses and 1,588 units are gone, with the bet that would recover them sitting at 1,747. Since three-quarters of seasons contain a six-loss run, the plan that requires the run to stop is the plan most likely to meet one. Flat stakes remove the mechanism entirely, which is one of the few unambiguous results on the bankroll page.
The deeper reason none of it helps is that a bettor can’t wager on the selection. Miller and Sanjurjo’s bias appears when somebody sifts a record after the fact, choosing which entries to count. A sportsbook offers one game at one number, chosen by them rather than by you, and the number carries no memory of Sunday. Streaks are real, the pattern in the record is real, and neither is available to buy.
Four losses in a row will keep feeling like something. Emotional discipline is largely the work of noticing that the feeling is well founded and that the arithmetic behind it is now published. It still won’t pay for the fifth bet.
Sources: Charles T. Clotfelter and Philip J. Cook, “The Gambler’s Fallacy in Lottery Play”, Management Science 39(12), 1993, pages 1521 to 1525. Joshua B. Miller and Adam Sanjurjo, “Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers”, Econometrica 86(6), 2018, pages 2019 to 2047. Thomas Gilovich, Robert Vallone and Amos Tversky, “The Hot Hand in Basketball”, Cognitive Psychology 17(3), 1985, pages 295 to 314. The Monte Carlo probability, the streak selection fractions, the run probabilities and the doubling schedule are computed from the stated assumptions.