One evening in 1913, at the casino in Monte Carlo, a roulette ball landed on black. Then again, and again. As the streak grew, players piled onto red, convinced it had to come up soon. The ball landed on black 26 times in a row. The people betting on red lost heavily, and the mistake they made now has a name: the gambler's fallacy, sometimes called the Monte Carlo fallacy.
The fallacy
The gambler's fallacy is the belief that past results of a random game change what happens next: that after a run of black, red is "due"; that a slot that hasn't paid in a while is about to; that a number that hasn't come up is more likely now.
A roulette wheel has no memory. On a European wheel, red has 18 pockets out of 37 on every spin. After one black, after ten, after twenty-five: the chance of red on the next spin is 18 in 37, about 48.6%. Always.
Isn't a long streak very unlikely?
Yes, before it starts. The chance of 26 blacks in a row on a fair European wheel is about 1 in 137 million. But by spin 26, the first 25 have already happened. The only question left is the next spin, and that's 18 in 37 like every other.
Streaks also aren't as rare as they feel. Wheels around the world spin millions of times a year, so long runs keep turning up somewhere. Shorter runs, like five or six of the same colour, happen at every table every night.
"It evens out in the long run"
It does, but not by correcting itself. Over thousands of spins, red and black each come up close to 18 in 37 of the time because the early streak gets diluted by everything that comes after it, not because the wheel pays it back. Nothing balances the books for the person who lost on the streak.
The same mistake in other games
- Slots: a machine that hasn't paid out isn't "due". Each spin is random and independent. See do slot machines get hot?
- Lottery: "overdue" numbers are no more likely than any others.
- Betting systems: doubling up until red comes is a gambler's fallacy with a plan. See why the Martingale doesn't work.
Whatever the last spin was, the house edge on the next one is the same. To see what it adds up to, try the roulette calculator.
Sources
- Medical Journal of Australia: The Monte Carlo fallacy (Owen, 2011)
- Wizard of Odds: Roulette basics