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
Sicherman dice dice is a mathematical exercise in elementary combinatorics, involving a re-labeling of the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling. The Sicherman dice are crazy dice that are re-labeled with only positive integers. The table below lists all possible totals of dice rolls with standard dice and Sicherman dice. One Sicherman die is coloured for clarity: 1–2–"2"–3–"3"–4, and the other is all black, 1–3–4–5–6–8. The Sicherman dice were discovered by George Sicherman of Buffalo, New York and were originally reported by Martin Gardner in a 1978 article in "Scientific