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Mathematics Question on Probability

Let A and B be two independent events such that P(A)=13P\left(A\right) = \frac{1}{3} and P(B)=16.P\left(B\right) = \frac{1}{6}. Then, which of the following is TRUE ?

A

P(A/(A∪B))=14P \left(A /\left(A \cup B \right)\right) = \frac{1}{4}

B

P(A/B′)=13P \left(A / B' \right) = \frac{1}{3}

C

P(A/B)=23P \left(A / B \right) = \frac{2}{3}

D

P(A′/B′)=13P \left(A' / B' \right) = \frac{1}{3}

Answer

P(A/B′)=13P \left(A / B' \right) = \frac{1}{3}

Explanation

Solution

(1) P(A/B)=P(A)13\left(1\right)\,P\left(A/ B\right)=P\left(A\right) \frac{1}{3}
(2) P(A/(A∪B))=P(A∩(A∪B))P(A∪B)=P(A)P(A∪B)\left(2\right)\,P\left(A/ \left(A\cup B\right)\right)=\frac{P\left(A\cap\left(A\cup B\right)\right)}{P\left(A\cup B\right)}=\frac{P\left(A\right)}{P\left(A\cup B\right)}
=1313+16−118=34=\frac{\frac{1}{3}}{\frac{1}{3}+\frac{1}{6}-\frac{1}{18}}=\frac{3}{4}
(3) P(A/B′)=P(A)=13\left(3\right)\,P\left(A/ B'\right)=P\left(A\right)=\frac{1}{3}
(4) P(A′/B′)=P(A′)=23\left(4\right)\,P\left(A'/ B'\right)=P\left(A'\right)=\frac{2}{3}