Gambler's Fallacy Part 1 - Casinos

The "gambler's fallacy" occurs in games of chance when a streak of one type of result happens, and the gambler then bets on the reverse. For example, supposed you were betting on coin flips and you waited until 10 heads in a row came up. You then bet on tails, because it's "due" to come up.

This is a fallacy because each flip in independent of one another. The chance of a tail coming up after 10 heads is still 50%. This same line of thinking is false in casino games such as baccarat or roulette. If a long string of black occurs, red is not more likely to come up.

Some gamblers will argue that, if there is no regression to the mean, how can the long term odds (50% in a coin toss) be so stable? In other words, say 100 tails in a row come up, how can the results regress back to 50/50 without a long streak of heads?

But you don't need a run of the other result. Suppose 100 tails in a row came up. If, as an example, the next 1000 flips alternate between head and tail, then the results become 600 tails, 500 heads. The odds have regressed toward the normal 50-50 without a corresponding run of heads.