Gambler's Fallacy - Other Examples

Other Examples

There is another way to emphasize the fallacy. As already mentioned, the fallacy is built on the notion that previous failures indicate an increased probability of success on subsequent attempts. This is, in fact, the inverse of what actually happens, even on a fair chance of a successful event, given a set number of iterations. Assume a fair 16-sided die, where a win is defined as rolling a 1. Assume a player is given 16 rolls to obtain at least one win (1−p(rolling no ones)). The low winning odds are just to make the change in probability more noticeable. The probability of having at least one win in the 16 rolls is:

However, assume now that the first roll was a loss (93.75% chance of that, 15⁄₁₆). The player now only has 15 rolls left and, according to the fallacy, should have a higher chance of winning since one loss has occurred. His chances of having at least one win are now:

Simply by losing one toss the player's probability of winning dropped by 2 percentage points. By the time this reaches 5 losses (11 rolls left), his probability of winning on one of the remaining rolls will have dropped to ~50%. The player's odds for at least one win in those 16 rolls has not increased given a series of losses; his odds have decreased because he has fewer iterations left to win. In other words, the previous losses in no way contribute to the odds of the remaining attempts, but there are fewer remaining attempts to gain a win, which results in a lower probability of obtaining it.

The player becomes more likely to lose in a set number of iterations as he fails to win, and eventually his probability of winning will again equal the probability of winning a single toss, when only one toss is left: 6.25% in this instance.

Some lottery players will choose the same numbers every time, or intentionally change their numbers, but both are equally likely to win any individual lottery draw. Copying the numbers that won the previous lottery draw gives an equal probability, although a rational gambler might attempt to predict other players' choices and then deliberately avoid these numbers. Low numbers (below 31 and especially below 12) are popular because people play birthdays as their so-called lucky numbers; hence a win in which these numbers are over-represented is more likely to result in a shared payout.

A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an aircraft, a man decides to always bring a bomb with him. "The chances of an aircraft having a bomb on it are very small," he reasons, "and certainly the chances of having two are almost none!" A similar example is in the book The World According to Garp when the hero Garp decides to buy a house a moment after a small plane crashes into it, reasoning that the chances of another plane hitting the house have just dropped to zero.