Suppose you’re playing Langer and Roth’s (1975) coin toss game with a fair coin (which you pulled out of your own pocket) and you are trying to predict the next outcome, heads or tails, after the coin has been tossed 8 times.
Remarkably, the coin has come up heads on each toss, a run of 8 heads.
If you’re like most people, you’ll have a feeling that tails is more likely on the ninth toss ― you feel “it’s due” ― and you’d probably even bet some money on the prediction of tails.
Another example of this feeling is the common, but incorrect advice about how to gamble: “When you’re in Vegas and you see a roulette wheel come up with a run of three or more reds, bet black.
You’re sure to win.”
There is even a rationale for this belief: Nine heads (or reds) in a row is very rare; the odds are strongly against this happening (1/2 or 1/512 or approximately .002 for the coin, less for the roulette wheel), so if you’re looking at 8 in a row, it’s very unlikely you’ll get 9 in a row.
This intuition and rationale are an error called the gambler’s fallacy ─ the notion that “chances of independent, random events mature” if they have not occurred for a while.
Fair coins and roulette wheels have no memories; the chance of each event is independent of all the other events in a sequence, and the probability of tails or red is constant.