## Probability Games - Dice Game

The game is based on throwing two dice (”a pair of dice”) simultaneously. In each round, you win a sum of money equal to the sum of the pair of dice.

For example, if you throw 3 with one of the dice and 4 with the other, then you will win $7. We will calculate the maximum sum that it is worthwhile paying for each round.

As we have already seen, when throwing a pair of dice, it is possible to receive 36 different results.

1 – 1 | 1 – 2 | 1 – 3 | 1 – 4 | 1 – 5 | 1 – 6 |

2 – 1 | 2 – 2 | 2 – 3 | 2 – 4 | 2 – 5 | 2 – 6 |

3 – 1 | 3 – 2 | 3 – 3 | 3 – 4 | 3 – 5 | 3 – 6 |

4 – 1 | 4 – 2 | 4 – 3 | 4 – 4 | 4 – 5 | 4 – 6 |

5 – 1 | 5 – 2 | 5 – 3 | 5 – 4 | 5 – 5 | 5 – 6 |

6 – 1 | 6 – 2 | 6 – 3 | 6 – 4 | 6 – 5 | 6 – 6 |

Sorting the Pairs According to the Sum of the Dice in Increasing Order

In the following table, we have listed all the pairs in groups.

Each group contains pairs with the same sum (the sum appears in Row 1).

Row 2 lists all of the pairs giving the sums listed in Row 1. Row 3 shows the number of such pairs in the column.

For example, there are 4 pairs with a sum of 5. Row 4 shows the probability, which is actually the number appearing in Row 3 (the size of the event), which is divided by 36 (the size of the sample space).

Row 1 | Sum of the Pairs | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 |

Row 2 | The pairs that give the sum in Row 1 | 4,3 | ||||||||||

4,4 | 3,4 | 3,3 | ||||||||||

5,4 | 5,3 | 5,2 | 4,2 | 3,2 | ||||||||

5,5 | 4,5 | 3,5 | 2,5 | 2,4 | 2,3 | 2,2 | ||||||

6,5 | 6,4 | 6,3 | 6,2 | 6,1 | 5,1 | 4,1 | 3,1 | 2,1 | ||||

6,6 | 5,6 | 4,6 | 3,6 | 2,6 | 1,6 | 1,5 | 1,4 | 1,3 | 1,2 | 1,1 | ||

Row 3 | Total number of pairs | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |

Row 4 | Probability | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |

Probability Distribution Table - Results

The Probability Distribution Table has three columns:

- The value (the sum of the dice, as it appears in Row 1 of the previous table).
- The theoretical probability.
- The contribution of each value to the expectation (column 1 multiplied by column 2).

The Value | The Probability | The Contribution to the Expectation |

2 | 1/36 | 2/36 |

3 | 2/36 | 6/36 |

4 | 3/36 | 12/36 |

5 | 4/36 | 20/36 |

6 | 5/36 | 30/36 |

7 | 6/36 | 42/36 |

8 | 5/36 | 40/36 |

9 | 4/36 | 36/36 |

10 | 3/36 | 30/36 |

11 | 2/36 | 22/36 |

12 | 1/36 | 12/36 |

Total | 1 | 252/36 = 7 |

The expectation is 7.

This means that were the trial to behave according to the theoretical probability, we would receive $7 on average in each round.

The maximum sum that we would be willing to pay for each round in order to make a profit is $6.99.