Craps principle and formula_29 ; in particular, the probability of a draw is irrelevant. If the game is played repeatedly until someone wins, then the conditional probability above is the probability that the player wins the game. This is illustrated below for the original game of craps, using an alternative proof. If the game being played is craps, then this principle can greatly simplify the computation of the probability of winning in a certain scenario. Specifically, if the first roll is a 4, 5, 6, 8, 9, or 10, then the dice are repeatedly re-rolled until one of two events occurs: Since
Craps gambler's fallacy indicates that a craps player should bet on eleven if an eleven has not appeared or has appeared too often in the last 20 rolls. In practice this can be observed as players respond to a roll such as a Hard Six with an immediate wager on the Hard Six. In reality, each roll of the dice is an independent event, so the probability of rolling eleven is exactly 1/18 on every roll, regardless of the number of times eleven has come up in the last x rolls. Even if the dice are actually biased toward particular results