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

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

Answer 1

The correct option is C $\frac{1}{4}$

$P\left(\overline{A}\right)=3/10\Rightarrow P\left(A\right)=7/10$

$P\left(B\right)=2/5\Rightarrow P\left(B\right)=3/5$

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

$=7/10+3/5-1/2$

$=4/5$

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

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

$P(B\cup \overline{B})=1\Rightarrow P(B\cup A\cup \overline{B})=1$

$\therefore P(B\cap (A\cup B\left)\right)=2/5+4/5-1$

$=1/5$

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