In probability theory, the craps principle is a theorem about event probabilities under repeated iid denote two mutually exclusive equals the conditional probability occur on the next trial, which is

need not be collectively exhaustive (if they are, the result is trivial).

Proof

are mutually exclusive and collectively exhaustive for the first trial, we have

gives the formula

1. .

Application

If the trials are repetitions of a game between two players, and the events are

then the craps principle gives the respective conditional probabilities of each player winning a certain repetition, given that someone wins (i.e., given that a draw ; in particular, the probability of a draw is irrelevant.

Stopping

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.

Craps example

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:

are mutually exclusive, the craps principle applies. For example, if the original roll was a 4, then the probability of winning is

This avoids having to sum the infinite series corresponding to all the possible outcomes:

ties followed by rolling the point:

The summation becomes an infinite geometric series:

which agrees with the earlier result.

References

Notes