Gambler's fallacy (definition and example)

The gambler's fallacy is the mistaken belief that, after a run of one result in a game of chance, the opposite result is now "due". It is wrong whenever the rounds are independent: the game has no memory, so the chance on the next round is exactly what it was before the run.

It is also called the Monte Carlo fallacy, after August 1913, when the roulette ball at the Monte Carlo Casino landed on black 26 times in a row and gamblers lost millions of francs betting against it.

A worked example with Coin Flip

In Cashout Arcade's Coin Flip you call heads or tails and climb a ladder of up to 15 flips. Suppose you have called five flips right.

• The chance of calling five in a row from the start was 1 in 32, or 3.125 percent. That is rare, which is why it feels like something must change.
• The chance the sixth flip goes your way is 50 percent, the same as flip one.

Nothing has been "used up". The 3.125 percent is the chance of the whole run before it began. Once five flips are behind you, only one flip is still uncertain.

The same logic works backward. Five tails in a row does not make heads more likely on flip six, and the coin does not know or care. The other face is not due.

Streaks are normal

Intuition says long streaks are rare, but in many trials they are routine. In 100 coin flips the chance of at least one run of 7 or more of the same face is about 54 percent, and the chance of a run of 6 or more is about 81 percent. A game without long runs would be less random, not more.

The related mirror error

The reverse mistake is the hot hand: believing a streak will continue. In independent games it will not, on average, for the same reason.

Common misconceptions

• "The law of averages will fix it." There is no force that corrects a streak. The long-run average improves because new results swamp old ones, not because old ones are cancelled. See Law of large numbers (casino example).
• "I have lost ten times, so I am due a win." Each round carries the same chance. A "due" win has no mathematical basis.
• "This only applies to coins." It applies to roulette, dice, slots and crash multipliers alike.

Believing a win is due is a common driver of chasing losses; see Chasing losses (definition and maths). For more examples see Gambler's Fallacy and Streak Myths in Casino Games. Coins in the app are virtual and have no cash value.