Let A and B be two events such that $P(A)=\frac{3}{8}, P(B)=\frac{5}{8}$ and $P(A∪B)=\frac{3}{4}$. Which of the following is correct ?
A. P(A|B) is $\frac{2}{5}$
B. P(A'|B) is $\frac{4}{5}$
C. P(A|B).P(A'|B) is $\frac{8}{25}$
D. P(B'|A) is $\frac{1}{3}$
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 3
The correct answer is option (3) → A and D only
$P(A∩B)=P(A∪B)+P(A)+P(B)$
$=-\frac{3}{4}+\frac{3}{8}+\frac{5}{8}=\frac{-6+3+5}{8}=\frac{1}{4}$
(A) $P(A|B)=\frac{P(A∩B)}{P(B)}=\frac{2}{5}$ True
(B) $P(\overline A|B)=\frac{P(\overline A∩B)}{P(B)}=\frac{P(B)-P(A∩B)}{P(B)}=\frac{2}{5}$ False
(C) $P(A|B).P(\overline A|B)=\frac{4}{25}$ False
(D) $P(\overline B|A)=\frac{P(A)-P(A∩B)}{P(A)}=\frac{\frac{3}{8}-\frac{1}{4}}{\frac{1}{4}}=\frac{1}{3}$ True
A and D → correct