Problem 48E

Galileo’s Passedix game. Passedix is a game of chance played with three fair dice. Players bet whether the sum of the faces shown on the dice will be above or below 10. During the late sixteenth century, the astronomer and mathematician Galileo Galilei was asked by the Grand Duke of Tuscany to explain why “the chance of throwing a total of 9 with three fair dice was less than that of throwing a total of 10” (Interstat, Jan. 2004). The grand duke believed that the chance should be the same because “there are an equal number of partitions of the numbers 9 and 10.” Find the flaw in the grand duke’s reasoning and answer the question posed to Galileo.

Answer

Step 1 of 1

We are asked to find the reason that the chance of throwing a total of 9 with three fair dice was less than that of throwing a total of 10.

The grand duke believed that the chance should be the same because “ there are an equal number of partitions of the number 9 and 10.”

We need to find the flaw in the grand duke’s reasoning and answer the right reasoning.

Since we have given the three fair dice, then there are 6 possible outcomes of each dice.

Hence the possible combinations in the sample space would be

Lets denote the first dice as the second dice as and the Third dice as

Lets form the sample space of getting a total of by three dice.

Similarly, getting total of 10 by three dice,

Since the grand duke believed there are an equal number of partitions of the number 9 and 10.

However, if we take into account the three identities (A, B, and C) of the dice, then there are various ways to get each partition.

For instance, to get a partition of 126, we will have combinations like (126, 162, 216, 261, 612, 621).

Similarly, we can think of a number of ways for each partition.

The chance of throwing a total of 9 |
Number of ways |

126 |
6 |

135 |
6 |

144 |
3 |

225 |
3 |

234 |
6 |

333 |
1 |

Total |
25 |

Similarly, a number of ways of getting a total of 10 for each partition.

The chance of throwing a total of 10 |
Number of ways |

136 |
6 |

145 |
6 |

244 |
3 |

226 |
3 |

235 |
6 |

334 |
3 |

Total |
27 |

Therefore, there is a total of 25 ways of getting a sum of 9 and 27 ways of getting a sum of 10.

Thus, the chance of throwing a total of 9 is less than the chance of throwing a total of 10.

Hence there is a flaw in the grand duke’s reasoning.