Question

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

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{8}$
• D. $\frac{2}{9}$

Answer 1

The correct option is B $\frac{2}{3}$

Given $P(A\cup B)=\frac{3}{4},P(A\cap B)=\frac{1}{4}$

$P\left(A\right)=1-P\left(\overline{A}\right)=\frac{1}{3}$

$⟹P\left(A\right)+P\left(B\right)-P(A\cap B)=P(A\cup B)=\frac{3}{4}⟹\frac{1}{3}+P\left(B\right)-\frac{1}{4}=\frac{3}{4}⟹\frac{1}{3}+P\left(B\right)=1⟹P\left(B\right)=\frac{2}{3}$