Two dice are rolled. Find the probability of the following events:(a) Event A: the product of the numbers on the upper faces is 12.(b) Event B: both dice show the same number.(c) Event C: the sum of the numbers is a prime number.

Answer: \(P(A) = \dfrac{1}{9}\), \(P(B) = \dfrac{1}{6}\), \(P(C) = \dfrac{5}{12}\)

n(S) = 36

(a) A = {(2,6), (6,2), (3,4), (4,3)}, n(A) = 4, \(P(A) = \dfrac{4}{36} = \dfrac{1}{9}\)

(b) B = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}, n(B) = 6, \(P(B) = \dfrac{6}{36} = \dfrac{1}{6}\)

(c) Possible prime sums are 2, 3, 5, 7, 11.

Sum 2: (1,1). Sum 3: (1,2), (2,1). Sum 5: (1,4), (2,3), (3,2), (4,1). Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Sum 11: (5,6), (6,5).

n(C) = 1 + 2 + 4 + 6 + 2 = 15, \(P(C) = \dfrac{15}{36} = \dfrac{5}{12}\)