Question

If two events $A$ and $B$ are such that $P\left(\overline{A}\right)=\frac{3}{10},P\left(B\right)=\frac{2}{5},$ and $P(A\cap \overline{B})=\frac{1}{2},$ then $P\left(\frac{B}{A\cup \overline{B}}\right)=$

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

Answer 1

The correct option is C

$\frac{1}{4}$

$P\left(\frac{B}{A\cup \overline{B}}\right)=\frac{P(B\cap (A\cup \overline{B}\left)\right)}{P(A\cup \overline{B})}=\frac{P(A\cap B)}{P(A\cup \overline{B})}$

Now, $P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)$

$\Rightarrow \frac{1}{2}=\frac{7}{10}-P(A\cap B)\left[P\right(A)=1-P(\overline{A}\left)\right]$

$\therefore P(A\cap B)=\frac{1}{5}$

Also, $P(A\cup \overline{B})=P\left(A\right)+P\left(\overline{B}\right)-P(A\cap \overline{B})$

$=\frac{7}{10}+\frac{3}{5}-\frac{1}{2}$

$=\frac{4}{5}$

$\therefore P\left(\frac{B}{A\cup \overline{B}}\right)=\frac{\frac{1}{5}}{\frac{4}{5}}=\frac{1}{4}$