Suppose A and B are events of a sample space S with P(A) = 0.42, P(B) = 0.34, and P (A or B) = 0.53.

Question

How do I find the probability of B and the complement of A?

Answer 1

X = Prob( A AND B)

Then Prob (A only) = 0.42 - x

Prob(B only) = 0.34 - x

Prob (A OR B) = 0.53 = 0.42 - x + 0.34 - x + x

0.53 = 0.42 + 0.34 - x

0.53 = 0.76 - x

x = 0.23 = Prob ( A and B)

Prob (A only) = 0.42-0.23 = 0.19

Prob (B only) = 0.34 - 0.23 = 0.11

Altogether = 0.11 + 0.19 + 0.23 = 0.53 = Prob (A or B)

So that leaves 0.47 in neither A nor B.

So Prob( not A and B) = Prob (B only) = 0.11 <--- not in A and in B = Universal set intersect B only