An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: \(A:\) the sum is greater than \(8\), \(B:\)\(2\) occurs on either die \(C:\)The sum is at least \(7\), and a multiple of \(3\). Which pairs of these events are mutually exclusive?

Question

For two events A and B, if \(P(A) = P(A/B) = \frac{1}{4}\) and \(P(B/A) = \frac{1}{2}\), then which of the following is not true?

Answer 1

Sample Space:

When a pair of dice is rolled, outcomes are ordered pairs (i, j) where
i, j ∈ {1, 2, 3, 4, 5, 6}

Event A: The sum is greater than 8

Possible sums: 9, 10, 11, 12

A = { (3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) }

Event B: 2 occurs on either die

B = { (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (1,2), (3,2), (4,2), (5,2), (6,2) }

Event C: The sum is at least 7 and a multiple of 3

Possible sums ≥ 7 and multiples of 3 are: 9 and 12

C = { (3,6), (4,5), (5,4), (6,3), (6,6) }

Checking mutually exclusive pairs:

A and B:

A ∩ B = { (4,5)? no 2, (5,4)? no 2, (3,6)? no 2, (6,3)? no 2, (6,6)? no 2 }

Since no outcome contains 2,
A ∩ B = ∅

Hence, A and B are mutually exclusive.

A and C:

C ⊂ A

A ∩ C ≠ ∅

Hence, A and C are not mutually exclusive.

B and C:

No outcome in C contains the number 2.

B ∩ C = ∅

Hence, B and C are mutually exclusive.

Final Answer:

The mutually exclusive pairs are:
(A, B) and (B, C)

Solution and Explanation