Question

If $P(A\cup B)=\frac{3}{4}$ and $P\left(\overline{A}\right)=\frac{2}{3}$, then find the value of $P(\overline{A}\cap B)$.

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{5}{12}$
• D. None of these

Answer 1

Answer: (C)

$\frac{5}{12}$

Since $\overline{A}\cap B$ and A are mutually exclusive events such that
$A\cup B=(\overline{A}\cap B)\cup A$
or $P(A\cup B)=P(\overline{A}\cap B)+P\left(A\right)$
or $\frac{3}{4}=P(\overline{A}\cap B)+1-\frac{2}{3}$
or $P(\overline{A}\cap B)=\frac{5}{12}$