Question

A pair of dice is rolled in a remote location and when you ask an honest observer whether at least one die came up six, this honest observer answers in the affirmative. a) What is the probability that the sum of the numbers that came up on the two dice is seven, given the information provided by the honest observer? b) Suppose that the honest observer tells us that at least one die came up five. What is the probability the sum of the numbers that came up on the dice is seven, given this information?

Answer 1

The probability of the sum being seven when at least one die is a six is 1/11, while the probability is 2/11 when at least one die is a five.

Step-by-step explanation:

The problem concerns the calculation of conditional probabilities when rolling a pair of dice. In part a), we are given that at least one die is a six, and we need to find the probability that the sum is seven. In part b), we are given at least one die is a five, and we need to find the probability that the sum is seven.

Part a)

If at least one die is a six, the possible outcomes are (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (1,6), (2,6), (3,6), (4,6), and (5,6), for a total of 11 outcomes. Only one of these, (6,1), results in a sum of seven. Therefore, the probability of the sum being seven is 1/11.

Part b)

If at least one die is a five, the possible outcomes are (5,1), (5,2), (5,3), (5,4), (5,6), (5,5), (1,5), (2,5), (3,5), (4,5), and (6,5), for a total of 11 outcomes. Two of these, (5,2) and (2,5), result in a sum of seven. Therefore, the probability of the sum being seven is 2/11.

Answer 2

(a) 2/11 = 0.1818, (b) 2/11 = 0.1818

Step-by-step explanation:

This is conditional probability problem as we are given conditions that one of the die must be 6 or 5 and the overall sum must be 7.

I have solved this problem on paper (Attached Herewith)