Gambler's fallacy
The gambler's fallacy is the mistaken belief that a run of one outcome makes the opposite outcome more likely next, even though independent trials keep the same probability every time.
Also known asgamblers-fallacy
What it is
See it move
A fair coin lands heads five times in a row. Calculated in advance, that whole streak had probability 0.5^5 = 3.125%. But each flip is independent, so on flip six the probability of heads is still exactly 0.5, or 50% — unchanged by the streak that already happened.
The formula
Variables
- Fixed probability of the outcome on a single independent trial
- Length of the streak (trials)
The probability of an entire streak, calculated in advance; it does not change the probability of any single trial within or after the streak, which stays at p.
Check yourself
On a fair roulette wheel (treat red/black as an even 50/50 split), the ball has landed on red 7 times in a row. A gambler believes black is now more likely on the next spin. If each spin is independent with P(black) = 0.5, what is the actual probability black appears on the 8th spin, and what was the probability, calculated in advance, of red appearing on all 7 spins?