Question

A pair of fair dice is rolled. Let E denote the event that the number falling uppermost on the first die is 2, and let F denote the event that the sum of the numbers falling uppermost is 8. (Round your answers to three decimal places.)
(a) Compute P(F).
(b) Compute P(E ∩ F).
(c) Compute P(F | E).
(d) Compute P(E).

Answer 1

First of all, note that there are 36 possible outcomes (all the numbers from 1 to 6 from the first die, combined with all the numbers from 1 to 6 from the second die).

a. Out of these 36 outcomes, the only ways to get 8 as sum are: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). There are 5 favourable cases over 36 possible cases, so the probability is 5/36.

b. We want to get 8 as sum, and the first die must be a 2. Out of the five pairs we got before, only the first satisfies this request, so the probability is 1/36

c. Knowing that the first die was a 2, we have 6 possible outcomes:

The second die rolls a 1, for a total of 3.

The second die rolls a 2, for a total of 4.

The second die rolls a 3, for a total of 5.

The second die rolls a 4, for a total of 6.

The second die rolls a 5, for a total of 7.

The second die rolls a 6, for a total of 8.

So, there is 1 favourable outcome out of 6 possible outcomes, so the probability is 1/6

d. The die is fair, so every number occurs with probability 1/6.