2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | W1 | L1 | P1 | W | L | Rolls |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 2 | 4 | 7 | 4 | 6 | 2 | 6 | 2 | 1 | 6 | 3 | 27 | 14 | 22 | 123 |

0 | 4 | 2 | 2 | 4 | 9 | 6 | 3 | 2 | 2 | 2 | 11 | 6 | 19 | 18 | 18 | 127 |

0 | 2 | 2 | 5 | 10 | 5 | 1 | 5 | 2 | 3 | 1 | 8 | 3 | 25 | 17 | 19 | 169 |

1 | 1 | 3 | 5 | 2 | 6 | 7 | 9 | 1 | 0 | 1 | 6 | 3 | 27 | 16 | 20 | 117 |

0 | 1 | 1 | 6 | 9 | 4 | 5 | 4 | 3 | 1 | 2 | 5 | 3 | 28 | 17 | 19 | 128 |

1 | 3 | 4 | 5 | 4 | 4 | 5 | 7 | 1 | 1 | 1 | 5 | 5 | 26 | 13 | 23 | 112 |

1 | 1 | 6 | 6 | 6 | 2 | 6 | 0 | 4 | 3 | 1 | 5 | 3 | 28 | 18 | 18 | 151 |

0 | 2 | 2 | 4 | 7 | 9 | 3 | 3 | 2 | 2 | 2 | 11 | 4 | 21 | 20 | 16 | 123 |

2 | 2 | 3 | 5 | 4 | 8 | 1 | 3 | 3 | 1 | 4 | 9 | 8 | 19 | 19 | 17 | 100 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | W1 | L1 | P1 | W | L | Rolls |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 1 | 5 | 4 | 8 | 1 | 6 | 4 | 3 | 0 | 11 | 4 | 21 | 20 | 16 | 100 |

1 | 2 | 3 | 5 | 5 | 4 | 8 | 2 | 2 | 4 | 0 | 8 | 3 | 25 | 17 | 19 | 107 |

1 | 4 | 3 | 3 | 5 | 5 | 3 | 7 | 1 | 3 | 1 | 8 | 6 | 22 | 16 | 20 | 94 |

2 | 2 | 2 | 4 | 3 | 9 | 6 | 3 | 3 | 1 | 1 | 10 | 5 | 21 | 14 | 22 | 126 |

0 | 2 | 4 | 5 | 3 | 8 | 4 | 7 | 3 | 0 | 0 | 8 | 2 | 26 | 16 | 20 | 141 |

0 | 0 | 5 | 2 | 4 | 5 | 8 | 2 | 2 | 5 | 3 | 10 | 3 | 23 | 17 | 19 | 130 |

0 | 1 | 4 | 5 | 4 | 4 | 4 | 3 | 7 | 2 | 2 | 6 | 3 | 27 | 16 | 20 | 100 |

0 | 1 | 1 | 4 | 7 | 6 | 3 | 6 | 3 | 4 | 1 | 10 | 2 | 24 | 15 | 21 | 124 |

1 | 1 | 1 | 5 | 6 | 10 | 1 | 3 | 3 | 4 | 1 | 14 | 3 | 19 | 24 | 12 | 114 |

0 | 0 | 5 | 3 | 4 | 10 | 5 | 3 | 5 | 0 | 1 | 10 | 1 | 25 | 23 | 13 | 156 |

3 | 4 | 3 | 2 | 6 | 8 | 3 | 1 | 3 | 2 | 1 | 10 | 8 | 18 | 21 | 15 | 87 |

Sect | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | W1 | L1 | P1 | W | L | Rolls |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6 | 17 | 25 | 42 | 53 | 51 | 40 | 36 | 24 | 15 | 15 | 66 | 38 | 220 | 152 | 172 | 1,150 |

2 | 9 | 20 | 32 | 43 | 51 | 77 | 46 | 43 | 36 | 28 | 11 | 105 | 40 | 251 | 199 | 197 | 1,279 |

Total | 15 | 37 | 57 | 85 | 104 | 128 | 86 | 79 | 60 | 43 | 26 | 171 | 78 | 471 | 351 | 369 | 2,429 |

Although hypothesis testing isn't covered until later, here are some things we'll be looking at later. Don't worry too much about what they mean for right now, but you may want to come back and look at them later.

Probability theory tells us that the chance of getting a sum of 2 is 1/36, the chance of getting a sum of 3 is 2/36, and so on. The sum of the dice should have a nice triangular pattern if the theoretical values hold true. The expected frequencies are found by multiplying the total number of games by the chance of getting a 2, 3, 4, ..., or 12.

This comparison can be visualized graphically in the chart. The bars represent the observed frequencies and the red line represents the expected expected frequencies. If the sides appear with equal frequency, then the sums should have the distribution shown by the red line. Another way of saying that is that the bars should be close to the red line at the center of the bars.

Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Observed | 15 | 37 | 57 | 85 | 104 | 128 | 86 | 79 | 60 | 43 | 26 | 720 |

Expected | 20 | 40 | 60 | 80 | 100 | 120 | 100 | 80 | 60 | 40 | 20 | 720 |

Chi-sq = 6.62833333, df = 10, p-value = 0.760005

The p-value represents the probability of getting the results we actually got if the sum of the dice have the distribution they're supposed to have. A p-value of less than 0.05 would cause us to question whether or not the dice were behaving as they were expected to behave. This could happen if the dice weren't properly rolled or if some team made up their results.

Another claim is that the probability of winning a game of craps is 244/495 and the probability of losing a game of craps is 251/495. We'll test the claim about the probabilty of winning a game of craps by comparing our results to the 244/495 and seeing whether or not our results are close enough that any differences could be do to chance alone or whether something unusual is going on (maybe our class is very lucky ... or unlucky ... at shooting craps. 244/495 is approximately 49.29% of the time.

Test of H_{0}: p = 244/495
vs H_{1}: p ≠ 244/495

x | n | sample p | 95% C.I. | Test statistic |
p-value | |
---|---|---|---|---|---|---|

lower | upper | |||||

351 | 720 | 0.487500 | 0.450990 | 0.524010 | -0.291 | 0.770749 |

The p-value is the chance of getting the results we obtained if the true chance of winning a game is 244/495 = 49.2323%. If the p-value is less than 0.05 (5%), then we consider our results to be too unusual to happen by chance alone decide that something must be going on.

Our winning rate of 48.75% appears to be close enough to the expected 49.29% to say that our results could have happened by chance alone. There doesn't appear to be anything unusual going on here.

Of course, if we did this again, we might get different results.

- 2-12
- The number of times that a 2, 3, 4, ..., 12 was the initial roll for the game of craps
- W1, L1, P1
- The number of times that the initial roll was a win (W1), loss (L1), or point (P1)
- W, L
- The number of times the overall result was a win (W) or loss (L)
- Rolls
- The total number of rolls necessary to play 36 games of craps