Question

Alice rolls a six faced fair die twice, and she obtains two numbers that we denote with X and Y , respectively. Fair die means that, in each roll, the six outcomes are equally likely. Let A be the event that X = 4, B be the event that X + Y = 6, and C be the event that Y = 5. (a) Write the sample space S for this experiment. (b) Are A and B independent? (c) Are A and C independent? (d) Are B and C independent? (e) Are A, B, and C independent? Why?

Answer 1

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

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

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

For independent events P(A and B) = P(A).P(B)

b) P(A and B) = 1/36

P(A) = 1/6, P(B) = 5/36

P(A and B) is not equal to P(A).P(B)

Therefore A and B are not independent

c)

P(A and C) = 1/36

P(A) = 1/6, P(C) = 1/6

P(A and C) =P(A).P(C)

Therefore A and C are independent

d)

P(C and B) = 1/36

P(C) = 1/6, P(B) = 5/36

P(C and B) is not equal to P(C).P(B)

Therefore C and B are not independent