Question

If two events $A$ and $B$ are such that $P\left(\overline{A}\right)=0.3,P\left(B\right)=0.5$ and $P(A\cap B)=0.3$, then $P\left(\frac{B}{A\cup \overline{B}}\right)$ is equal to

• A. $\frac{5}{8}$
• B. $\frac{7}{8}$
• C. $\frac{3}{8}$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

$\frac{3}{8}$

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

$=(1-P\left(\overline{A}\right))+(1-P\left(B\right))-(P\left(A\right)-P(A\cap B))=(1-0.3)+(1-0.5)-(0.7-0.3)=0.8$

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

$=\frac{P(A\cap B)}{P(A\cup \overline{B})}=\frac{0.3}{0.8}=\frac{3}{8}$