Assume that A1, A2, and B are three events on a sample space such that P(B) > 0.

Then,

(a) P(A1|B) = P(A1A2 ∪ A1A′

2|B)

(b) P(A1 − A2|B) = P(A′

1A′

2|B)

(c) P(A1A2|B) = 1 − P(A′

1A′

2|B)

(d) P(A1A2|B) = 1 − P(A′

1 ∪ A′

2|B)

(e) P(A′

1 ∪ A′

2|B) = P(A′

1|B) + P(A′

2|B)