Craps principle

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In probability theory, the craps principle is a theorem about event probabilities under repeated iid trials. Let and denote two mutually exclusive events which might occur on a given trial. Then the probability that occurs before equals the conditional probability that occurs given that or occur on the next trial, which is

The events and need not be collectively exhaustive (if they are, the result is trivial).[1][2]


Let be the event that occurs before . Let be the event that neither nor occurs on a given trial. Since , and are mutually exclusive and collectively exhaustive for the first trial, we have

and . Since the trials are i.i.d., we have . Using and solving the displayed equation for gives the formula



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 does not occur). In fact, the result is only affected by the relative marginal probabilities of winning and  ; 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.

Craps example[edit]

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 and 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:

Mathematically, we can express the probability of rolling ties followed by rolling the point:

The summation becomes an infinite geometric series:

which agrees with the earlier result.


  1. ^ Susan Holmes (1998-12-07). "The Craps principle 10/16". Retrieved 2016-03-17.
  2. ^ Jennifer Ouellette (31 August 2010). The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse. Penguin Publishing Group. pp. 50–. ISBN 978-1-101-45903-4.