Please Explain If A B and C are events and wc know that the

Please Explain

If A, B, and C are events, and wc know that the pair A and B are independent, and that Band Care independent, can we conclude that A and C are independent?

Solution

No, we cannot conclude this. Counter example: Consider tossing a dice and a coin

A = face of dice is even.

B = coin shows head

C = face of dice is odd

Clearly A and B are independent, B and C are also independent, but A and C are not independent. P(A) = 1/2, P(C) = 1/2, P(A and C) = 0 which is not equal to P(A) * P(C)