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# Probability of Winning Craps

**8/8/04**

Craps is one of the most exciting games in the casino! It has one of the best chances of winning (ie. your expected winnings are not too negative).

There are many superstitions in craps (in all games of chance for that matter) and people claim to have sure-fire systems of betting, of holding the dice, of what type of thrower is good luck for the table, of what numbers shouldn't even be uttered during game play, etc.. To me, only the probabilities embedded in the game are what really matter, as one cannot make an unfair game fair just by changing playing style. I thought I'd calculate the probability of winning craps for the first couple of rolls.

To play this game you need two dice.

If the sum of the tops of the dice is 7 or 11 on the come-out roll (the roll that starts the game), the thrower wins, and retains the dice for another throw. If the sum is 2, 3, or 12 on the come-out roll, the thrower loses but retains the dice.

If the sum is a number other than a 7, 11, 2, 3, or 12, say x (called the thrower's point), the thrower must keep throwing, and only wins when the sum is x. If a 7 occurs before x, then the thrower loses, and passes the dice to the next thrower.

The probability of winning on a throw is obviously conditional on what the thrower's first throw was. Let P(X=x) be the probability of rolling a sum of x, and P(win|X=x) be the probability of winning given that you rolled an x.

This table shows the probabilities involved:

X |
P(X=x) |
P(win|X=x) |
P(X=x)P(win|X=x) |

2 |
1/36 |
0 |
0 |

3 |
2/36 |
0 |
0 |

4 |
3/36 |
3/9 |
9/324 |

5 |
4/36 |
4/10 |
16/360 |

6 |
5/36 |
5/11 |
25/396 |

7 |
6/36 |
1 |
6/36 |

8 |
5/36 |
5/11 |
25/396 |

9 |
4/36 |
4/10 |
16/360 |

10 |
3/36 |
3/9 |
9/324 |

11 |
2/36 |
1 |
2/36 |

12 |
1/36 |
0 |
0 |

Note that summing the P(X=x)*P(win|X=x) gives P(win), which is 244/295 ~ 49.29%.

This probability is a little *worse* than a coin flip, and is where the casino,
among other places, gets a little edge over the player. Next, add in the scores of non-mathematical factors,
such as psychological factors, that are against you (flashy lights, distractions, betting, daring, showing off,
booze, etc.) and you are at more of a disadvantage.

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