Question

Example 8.29 If \( A \) and \( B \) are two events such that \( P(A)=\frac{1}{4}, P(B)=\frac{1}{2} \) and \( P(A \) and \( B)=\frac{1}{8} \), find (i) \( P(A \) or \( B \) ) (ii) \( P( \) not \( A \) and \( \operatorname{not} B) \).

Answer 1

We have P(A) = 1/4, P(B) = 1/2

and P(A and B) = P(A ∩ B) = 1/8

(i) P(A or B) = P(A ∪ B) = P(A)+P(B) - P(A ∩ B)

= 1/4 + 1/2 - 1/8

\(=\frac{2+4-1}{8}\) = 5/8

(ii) P(not A and not B) = P(A' ∩ B')

= P((A ∪ B)')  (∵ (A ∪ B)' = A' ∩ B')

= 1 - P(A ∪ B)  (∵ P(A') = 1 - P(A))

= 1 - 5/8

= 3/8.