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
Long dice Design advantages of long dice include being relatively easy to create fair dice with an odd number of faces, and (for four-faced dice) being easier to roll than tetrahedral d4 dice (as found in many role-playing games). Both cubic dice and four-faced long dice are found as early as the mid third millennium BCE at Indus Valley Civilisation sites; these are marked variously with dot-and-ring figures, linear devices, and Indus Valley signs. Dot-and-ring figures are used to this day on long dice in India, and predominate in the central European long dice shown above. In India, long dice ("pasa") are