The gambler’s fallacy is the belief that roulette wheels, dice, etc., have “memory,” so that, for example, having observed an unlikely streak of losses, the probability that the next outcome will be a win has increased, as a correction toward the expected long-term trend. The Wikipedia page provides a good example involving repeatedly flipping a fair coin:
“If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is [only] 1/32, a person might believe that the next flip would be more likely to come up tails rather than heads again. This is incorrect and is an example of the gambler’s fallacy.”
That is, having observed a streak of four heads in a row, we are actually just as likely to observe heads again on the subsequent fifth flip as we are to observe tails. Similarly, even after betting on red at the roulette wheel and losing four times in a row, we should still expect to win a fifth such bet on red the same stubborn 18/38 of the time (assuming a typical double-zero American wheel).
So, here is what I think is an interesting puzzle: let’s play a game. I will flip a fair coin times, and prior to each flip, if you have observed a current streak of or more consecutive heads, then make a note of the outcome of the subsequent flip. After all 100 coin flips, tally the noted “streak-following” flips: if there are more heads than tails, then I pay you one dollar. If there are more tails than heads, then you pay me one dollar. (If there are an equal number of heads and tails, then we push.)
If the gambler’s fallacy is indeed a fallacy, then shouldn’t this be a fair bet, i.e., with net zero expected return? But I claim that I have a significant advantage in this game, taking more than 15 cents from you on average every time we play! Following a streak of heads, we expect to observe a much larger proportion of “trend-correcting” tails than “streak-extending” heads.
And there is nothing special or tricky about this particular setup. Try this experiment with a different number of coin flips, or a longer or shorter “target” streak length , or even a roulette-like bias on the coin flip probability. Or instead of focusing only on streaks of consecutive heads (i.e., ignoring streaks of tails), look for streaks of either or more heads or or more tails, and note whether the subsequent flip is different. The effect persists: on average, we observe a larger-than-expected proportion of outcomes that tend to end the streak.