Question

If $A$ and $B$ are two events such that $P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3}$ and $P\left(\overline{B}\right)=\frac{1}{3}$, then the value of $P\left(A\right)$ is

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

Answer 1

Answer: (C)

$\frac{1}{2}$

Given, $P\left(\overline{B}\right)=\frac{1}{3}\Rightarrow P\left(B\right)=\frac{2}{3}$

$P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3}$

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

$\frac{5}{6}=P\left(A\right)+\frac{2}{3}-\frac{1}{3}$

$P\left(A\right)=\frac{5}{6}-\frac{2}{3}+\frac{1}{3}=\frac{1}{2}$