Question

Suppose you roll two dice. Assume one is red and one is blue so that they are distinguishable. Let A be the event "The sum of the two dice is 7." and B be the event "The red die is 4." and C be the event "The difference of the two dice is even."

a.) Are A and B independent? Justify your answer.

b.) Are B and C independent? Justify your answer.

c.) Are A and C independent? Justify your answer.

Answer 1

When 2 events A and B are independent, P(A) x P(B) = P(A and B)

Number of outcomes on roll of two dice = 6x6 = 36

P(A) = 6/36 {(6,1), (1,6), (2,5), (5,2), (3,4), (4,3)}

= 1/6

P(B) = P(red dies is 4)

= 6/36

= 1/6

P(C) = P(differences is even)

= P(both are odds) + P(both are evens)

= 3/6 x 3/6 + 3/6 x 3/6

= 1/2

a) P(A and B) = 1/36    {(4,3)}      

P(A) x P(B) = 1/6 x 1/6 = 1/36

P(A and B) = P(A) x P(B)

So, A and B are independent

b) P(B and C) = 3/36 {(4,2), (4,4), (4,6)}

= 1/12

P(B) x P(C) = 1/6 x 1/2

= 1/12

P(B and C) = P(B) x P(C)

So, B and C are independent

c) P(A and C) = 0

P(A) x P(C) $\small \neq$ 0

So, A and C are not independent