Let X and Y be two events such that P(X)=13, P(X/Y)=12 and P(Y/X)=25. Then,

# Let $X$ and $Y$ be two events such that  and $P\left(Y/X\right)=\frac{2}{5}.$ Then,

1. A

$P\left(Y\right)=\frac{7}{15}$

2. B

$P\left(X\cap Y\right)=\frac{1}{5}$  $P\left(X\cup Y\right)=\frac{2}{6}$

3. C

$P\left(X\cup Y\right)=\frac{2}{5}$

4. D

$P\left(Y\right)=\frac{4}{15}$

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### Solution:

We have,

$P\left(Y/X\right)=\frac{2}{5}⇒\frac{P\left(X\cap Y\right)}{P\left(X\right)}=\frac{2}{5}⇒P\left(X\cap Y\right)=\frac{2}{5}P\left(X\right)=\frac{2}{5}×\frac{1}{3}=\frac{2}{15}$

It is given that

$P\left(X/Y\right)=\frac{1}{2}⇒\frac{P\left(X\cap Y\right)}{P\left(Y\right)}=\frac{1}{2}⇒P\left(Y\right)=2P\left(X\cap Y\right)=2×\frac{2}{15}=\frac{4}{15}$

So, option $\left(d\right)$ is correct.

Thus, we have

and $P\left(X\cap Y\right)=\frac{2}{15}$

So,

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