Question

If $A$ and $B$ are two events such that $P(A\cup B)=\frac{3}{4},P(A\cap B)=\frac{1}{4},P\left(\overline{A}\right)=\frac{2}{3}$ where $\overline{A}$ is the complement of $A$, then what is $P\left(B\right)$ equal to?

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{9}$
• D. $\frac{2}{9}$

Answer 1

Answer: (B)

$\frac{2}{3}$

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)\Rightarrow P(A\cup B)=1-P\left(\stackrel{\_}{A}\right)+P\left(B\right)-P(A\cap B)\Rightarrow \frac{3}{4}=1-\frac{2}{3}+P\left(B\right)-\frac{1}{4}\Rightarrow P\left(B\right)=\frac{2}{3}$
Hence, option B is correct.