On August 18, 1913, a roulette wheel at the Monte Carlo Casino landed on black 26 consecutive times, odds of roughly one in 68.4 million. The crowd that gathered was sure red was due and pushed millions of francs onto it as the streak ran on. They lost fortunes to a belief that random events keep a ledger. This episode takes that night as the entry point to the gambler’s fallacy, using a simple coin toss to show the error: five heads in a row is a 1 in 32 shot before you start, but after four heads the fifth flip is still 50-50, because the coin has no memory.

The instinct is old. In 1796 Pierre-Simon Laplace described expectant fathers who feared that a run of boys born nearby had used up the supply. Amos Tversky and Daniel Kahneman traced it to a belief in the law of small numbers, and brain imaging shows what happens after a loss. The episode then follows the bias into asylum courts, ballparks, banks, lotteries, and video games, and asks whether it can be unlearned.

• U.S. asylum judges were 5.5 percent less likely to approve a request after granting the previous two, and loan officers were 8 percent less likely to approve after a prior approval.
• Major League Baseball umpires were 1.3 percent less likely to call a strike when the previous two pitches had been called strikes.
• When the number 244 won a lottery in April 1988, 41 players had picked it. Three days later only 24 did, and it took two months for bets to recover.
• A 1967 experiment found that people explicitly taught about the fallacy performed identically to those who were not.
• The fix that worked was reframing: when participants treated a toss as the first of a new block instead of the last of an old one, the fallacy disappeared.