Is it possible for A and B to be independent if A is a subse

Is it possible for A and B to be independent if A is a subset of B? If it is impossible, explain why; if it is possible, give an example.

Solution

This is not possible because if A is subset of B, P(B | A) = 1 because any even in A is also in B.

If we take B such that P(B)<1, then P(B) is not equal to P(B|A), i.e, probability of B changes if A occurs. Thus it is impossible for A and B to be independent.