5. A pair of fair dice are rolled and the sum of the numbers on the top faces is noted.

- a) Write out a probabilistic model for this experiment.
- b) What is the probability that the number is even?
- c) What is the probability that the number is divisible by 5?
- d) What is the probability that the number is divisible by 5 given that it is even?
- e) Are being even and being divisible by 5 independent events?

a) There are two probabilistic models that we have used for the experiment of rolling a pair of dice. The first one is the 36 outcome equally likely probabilistic model which is based on the fact that if the dice are fair, then any face is as likely to come up as any other face on each die. The multiplication principle says that since there are 6 faces on each die, there are 36 total possible outcomes. They can be listed in a table as

This model can be streamlined a bit in this problems because we are only interested in the sum of the number on the top faces, not the individual numbers making up the sum. We see from the body of the table that the outcomes can be sumarized as the numbers between 2 and 12, and the probability of each number is the number of times that number appears in the table divided by 36.

n |
p |

2 |
1/36 |

3 |
2/36 |

4 |
3/36 |

5 |
4/36 |

6 |
5/36 |

7 |
6/36 |

8 |
5/36 |

9 |
4/36 |

10 |
3/36 |

11 |
2/36 |

12 |
1/36 |

totals |
36/36 |

b) To compute the probability that the number is even, add up all the probabilities of all the outcomes in the event

c) To compute the probability that the number is divisible by 5, add up all the probabilities of all the outcomes in the event

d) To get the conditional probability that it is divisible by 5 given that it is even, we use the formula for the definition of conditional probability.

In this case, the intersection of E and F is {10}, so we get

e) 1/6 is just slightly less than 7/36, so the events are not independent. If we know that the number is even, that makes it a little less likely that the numbedr will be divisible by 5.