Probability 3 Question 21
21. Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$. Then
(2017 Adv.)
(a) $P(Y)=\frac{4}{15}$
(b) $P\left(X^{\prime} Y\right)=\frac{1}{2}$
(c) $P(X \cup Y)=\frac{2}{5}$
(d) $P(X \cap Y)=\frac{1}{5}$
Show Answer
Solution:
- $P(X)=\frac{1}{3}$
$$ \begin{aligned} & P \frac{X}{Y}=\frac{P(X \cap Y)}{P(Y)}=\frac{1}{2} \\ & P \frac{Y}{X}=\frac{P(X \cap Y)}{P(X)}=\frac{2}{5} \end{aligned} $$
$P(X \cap Y)=\frac{2}{15}$
$$ P(Y)=\frac{4}{15} $$
$$ P \frac{X^{\prime}}{Y}=\frac{P(Y)-P(X \cap Y)}{P(Y)}=\frac{\frac{4}{15}-\frac{2}{15}}{\frac{4}{15}}=\frac{1}{2} $$
$P(X \cup Y)=\frac{1}{3}+\frac{4}{15}-\frac{2}{15}=\frac{7}{15}=\frac{7}{15}$
