Question

P( A ∩ B ) = 0.12 ,    P( A^c∩ B ) = 0.18 ,   and A & B are independent events.

      What is   P( A^c ∪ B ) ?

Answer 1

We are given here that:
P( A ∩ B ) = 0.12 which means that P(A and B) = 0.12

Also, we are given here that:
P( A^c∩ B ) = 0.18, which means that P( not A and B) = 0.18

Therefore, P(B) = P( A ∩ B ) + P( A^c∩ B ) = 0.18 + 0.12 = 0.3.

As A and B are independent events, therefore,
P(A) = P( A ∩ B ) / P(B) = 0.12 / 0.3 = 0.4

P(A^c) = 1 - P(A) = 1 - 0.4 = 0.6

Using law of addition of probability, we have here:
P(A^c   ∪ B) = P(A^c) + P(B) - P( A^c∩ B ) = 0.6 + 0.3 - 0.18 = 0.72

Therefore 0.72 is the required probability here.