Question

If A and B are two events such that $P\left(A\right)=\frac{3}{8},$ $P\left(B\right)=\frac{5}{8}$ and $P\left(A\cup B\right)=\frac{3}{4},$ then $P\left(\frac{B}{A}\right)=$

• A. $\frac{2}{3}$
• B. $\frac{3}{5}$
• C. $\frac{4}{5}$
• D. $\frac{1}{5}$

Answer 1

Answer: (A)

$\frac{2}{3}$

Given, $P\left(A\right)=\frac{3}{8},$......(1) $P\left(B\right)=\frac{5}{8}$ and $P\left(A\cup B\right)=\frac{3}{4}.$

Now we've,

$P[A\cup B]$$=P\left[A\right]+P\left[B\right]-P[A\cap B]$

or, $P[A\cap B]$$=P\left[A\right]+P\left[B\right]-P[A\cup B]$

or, $P[A\cap B]$$=\frac{3}{8}+\frac{5}{8}-\frac{3}{4}$

or, $P[A\cap B]=\frac{1}{4}$.......(2)

Now,

$P\left[\frac{B}{A}\right]$

$=\frac{P[A\cap B]}{P\left[A\right]}$

$=\frac{\frac{1}{4}}{\frac{3}{8}}$ [ Using [1] and [2] ]

$=\frac{2}{3}$.