The gambler’s fallacy is the mistaken belief that if you get a lucky streak, the odds are that in the long run your chances will even out. It also works in reverse with an unlucky streak. A simple example: you toss a coin 10 times, hoping to get heads. Logic tells you that if you toss the coin 10 times, 5 tosses will result in tails and 5 will result in heads. The first five coin tosses are heads. You mistakenly think that the next five tosses will probably be five tails.

You toss a fair coin 10 times in a row and get 10 heads. If you continue to toss the coin until you’ve tossed it a hundred times, will the heads and tails even out so you end up with roughly 50 tails and 50 heads?

If you answered yes, then you’ve just fallen into the gambler’s fallacy. If you toss a coin 10 times and get all heads, that uncommon steak isn’t going to be evened out by a streak of tails in the future.

A fallacy is a mistaken belief, usually a belief based on a faulty argument. The reasoning or argument in answering “yes” to the above question is that given a fair coin, if you toss it enough times, you’ll come up with 50 percent heads and 50 percent tails.* And usually tossing it a hundred times is enough to even out the heads/tails. However, a streak of heads (or tails) is very unusual. In fact, you have the tiny probability of 00.098% of that happening. Now consider what happens on the 13th coin toss: you have a 50 percent chance of a heads or a tails. And so for the 14th toss, and the 15th toss. So you’re likely to end up with (for the next 90 tosses) 45 heads and 45 tails, giving you a total of 55 heads and 45 tails.

A second example of the gambler’s fallacy; you play an online game with one of your friends and over the last year you’ve won 50% of your games. Your friend has a winning streak of 4 games in a row. You mistakenly think that the next 4 games will probably be wins for you as you are “due” for a win.

The reversal of the Gambler’s fallacy is also a fallacy, where (because the gambler thinks they are on a “lucky streak”) that they are more likely to get the same result (in the first example above, more tails).

Gambler’s fallacy is also known as the Monte Carlo Fallacy or the fallacy of maturity of chances.

A joke tells the tale of the man stopped from boarding an airplane when he was found to be carrying a bomb. When questioned as to why he was taking a bomb, he reasons, “The chances of an airplane having a bomb on it are very small, and certainly the chances of having two are almost none!” Similarly, the hero in The World According to Garp buys a house that has just had a plane crash into it. His reasoning is that the chances of another plane crashing into it are practically zero.

Or how about holding a lightning rod in a storm. You’re chances of getting hit by lightning are pretty high. If you get hit once (and survive), are you going to keep on holding the lightning rod? Probably not. Technically speaking, as long as the outcomes are independent events, the probability is exactly 50 percent.

Reference: Blog post by Vincent Granville in Data Science Central.