Question

Suppose A and B are events such that P(A) = 0.54, P(B) = 0.38, and P(A ∩ B) = 0.24.

(a)   Compute: (i) P(A ∩ B^c)   (ii) P(A ∪ B)   (iii) P(A^c ∪ B)   (iv) P(A^c | B^c)

(b)   Are events A and B disjoint? Justify your answer.

Answer 1

1)

P(A ∩ B^c) will have only A elements that are not part of the intersection

P(Only A)= P(A)-P(A ∩ B)= 0.54-0.24= 0.3

2)

We have,

P(A ∪ B)=P(A)+P(B)-P(A ∩ B)= 0.54+0.38-0.24= 0.68

3)

P(A^c ∪ B) will have the following elements-

=1-P(A)+P(B)-P(A^c ∩ B)

P(A^c ∩ B) will have elements that are part of only B

=1-0.54+0.38-0.14

=0.7

4)

P(A^c | B^c)= P(A^c ∩ B^c)/P(B^c)

=P(Φ)/(1-P(B))

We have P(Φ)= 1- {P(Only A)+P(Only B)+P(A ∩ B)}

=1-{0.3+0.14+0.24}=0.32

We have 0.32/(1-0.38)= 0.516