In probability theory, the craps principle is a theorem about event probabilities under repeated iid trials. Let
Contents
The events
Proof
Let
and
The other equation follows from the definition of conditional probability and the fact that
and
so by the definition of conditional probability,
Combining these three yields the desired result.
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 does not occur). In fact, the result is only affected by the relative marginal probabilities of winning
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.
Etymology
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
This avoids having to sum the infinite series corresponding to all the possible outcomes:
Mathematically, we can express the probability of rolling
The summation becomes an infinite geometric series:
which agrees with the earlier result.