Question

Question 5: You are given a fair red die and a fair blue die, and roll both dice uniformly at random and independently of each other. Define the events A = “the sum of the results is at least 9”, B = “at least one of the two rolls results in a 2”, C = “at least one of the two rolls results in a 5”. • Determine Pr(A), Pr(B), and Pr(C). • Determine Pr(B | C). • Are the events A and B independent? • Are the events A and C independent?

Question 5: You are given a fair red die and a fair blue die, andro both dice uniformly at random and independently of each other. Define the events A "the sum of the results is at least 9 B"at least one of the two rolls results in a 2", C"at least one of the two rolls results in a 5 Determine Pr(A), Pr(B), and Pr(C) Determine Pr(B |C) .Are the events A and B independent? Are the events A and C independent?

Answer 1

Possible outcomes are

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)

(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)

(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)

(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)

(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

P(A) = 10/36 = 5/18

P(B) = 11/36

P(C) = 11/36

P(B and C) = 2/36

P(B|C) = P(B and C)/P(C) = (2/36)/(11/36) = 2/11

P(A and B) = 0

As P(A) * P(B) not equals to P(A and B), events A and B are not independent.

P(A and C) = 5/36

As P(A) * P(C) not equals to P(A and C), events A and C are not independent.