The gambler's fallacy, explained

What it is

The gambler's fallacy is the mistaken belief that if something has happened more than usual recently, it will happen less often next time, or the reverse, in a process where each event is independent. A fair coin has a 50% chance of heads every flip, regardless of the last ten.

A famous example

On 18 August 1913, in a casino in Monte Carlo, the roulette ball fell on black twenty-six times in a row. Gamblers bet heavily on red, convinced it was overdue, and lost millions of francs. The wheel had no memory. Each spin was independent.

How it shows up in lotteries

• "I have played for three months and never won, so I am due."
• "That number has not appeared for ages, so it is bound to."
• "I almost won last time, so I should buy more."
• "I will keep playing until I win back what I lost."

Why it feels right

We expect small samples to look like the long run. Six heads in a row feels unbalanced, so we expect a correction. But in the long run there is no correction. Over millions of flips the proportion settles near half, because new flips dilute the early streak, not because the coin compensates.

The related mistake: the hot hand

The opposite error is believing a streak will continue ("I am on a roll"). In games of pure chance, streaks are just chance. Both errors treat random outcomes as if they had momentum.

How to protect yourself

1. 1Set your budget before you play and keep to it, whatever happens.
2. 2Treat every draw as a fresh start with the same odds.
3. 3Never increase stakes to win back a loss.
4. 4If you notice the thought "I am due", take it as a sign to stop for the week.

Seeing independence with numbers

Flip a fair coin ten times. The chance of ten heads in a row is about 0.1%. But if you have already flipped nine heads, the chance that the tenth is also heads is still exactly 50%. The rare event was the whole sequence, not the next flip. Confusing "this sequence is unlikely" with "the next flip is due" is the heart of the fallacy.

Lottery draws and independence

Each weekly draw uses a fresh ticket list and a fresh random value. Nothing from last week's draw carries over. A ticket you held last week does not get stronger or weaker. The fact that you have not won in ten draws does not change your chance in the eleventh.

Healthy ways to think instead

• Each draw is a separate, small, fun bet with a known cost.
• Measure success by whether you stayed within your budget, not by whether you won.
• If the thought "I am due" appears, take it as a cue to check your limits.

Example: the "due" number

A player notices that the number 23 has not appeared in a 6-from-49 draw for 40 draws. The chance it appears in any one draw is 6/49, about 12.2%. The chance it misses 40 draws in a row is 0.878 to the power 40, about 0.5%, so the streak is unusual. But with 49 numbers, some number will have a long gap, and the chance that 23 appears in the next draw is still 12.2%, exactly as it was after one draw. The gap does not store up pressure. Expecting 23 to be "due" is the fallacy.

Cues that you are thinking this way

When you catch one of these thoughts, stop and check your budget.

• "It has to come up soon."
• "I have lost so many times, my luck must change."
• "I'll keep playing until I win it back."
• "That was so close, I'll buy a few more."