## The 1,980 rolls.

The best source of information on Craps (or any gambling game) is the Wizard of Odds. However, it mostly for my own purposes, I reproduce here a table of all the possible outcomes of a craps hand because I find it useful to have handy! The “Rolls” is a the number of rolls out of 1980. Why 1980? Because that is the smallest integer you can use such that all the numbers in the table are integers too. (1980=36*55).

Event | Rolls |
---|---|

Craps 2 | 55 |

Craps 3 | 110 |

Point 4 or 10 win | 55 (x2) |

Point 4 or 10 lose | 110 (x2) |

Point 5 or 9 win | 88 (x2) |

Point 5 or 9 lose | 132 (x2) |

Point 6 or 8 win | 125 (x2) |

Point 6 or 8 lose | 150 (x2) |

7 win | 330 |

11 win | 110 |

Craps 12 | 55 |

Total Rolls | 1980 |

The sum of all the winning rolls is 976, and the sum of all the winning don’t pass is 949. So we can compute the house edge directly:

House edge pass = \(2*p-1 = 2*(976/1980)-1 = -7/495 = -0.0141414141.. \)

House edge don’t pass = 2*(949/(976+949))-1 = -27/1925 (about 0.01403)

Actually because the don’t pass pushes on the 12, there are two ways to compute the house edge: Per bet resolved as I have done it, or per bet made, i.e. considering the bet resolved on a 12. This later method is more common, and easier to use in later calculations:

2*(949/1980)+(55/1980)-1 = -3/220 = -0.01363636…

## Rolls per decision

It is also interesting and useful to know the average number of rolls per decision. We can compute this by taking the expectation value of the number of rolls give the expected number of rolls in each case.:

Number of rolls = 1 + prob(point is 4/10)*(number of rolls to resolve a 4/10) + prob(point is 5/9)*(number for 5/9)+ prob(point is 6/8)*(number for 6/8)

The probability of each point is easy (and in the table as well), but to compute the number of rolls to resolve each point we need the basic result that if an event has probability p, it takes on average 1/p trials for the event to occur. Given that we can see that the number of rolls is:

Point | number of rolls |
---|---|

4/10 | 36/9=4 |

5/9 | 36/10=3.6 |

6/8 | 36/11=3.272727… |

Putting that all together, the expected number of rolls is

1 + 1/6*4+2/9*3.6+5/18*3.2727…. = 3.375757……

You can use this to compute the expected number of decisions per hour if you know the number of rolls per hour. If there are about 120 rolls per hour, you would expect about 32 decisions per hour on your pass line bet.