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 $P\left(\overline{A}\right)=\frac{2}{3}$, where $\overline{A}$ is the complement of $A$, then what is the value of $P\left(B\right)$

• A. $1/3$
• B. $2/3$
• C. $1/9$
• D. $2/9$

Answer 1

The correct option is B $2/3$

Given for two events $A,B$

$P\left(\overline{A}\right)=\frac{2}{3}\Rightarrow P\left(A\right)=\frac{1}{3}$......(1).

Also given $P(A\cap B)=\frac{1}{4}$.......(2).

Also given,

$P(A\cup B)=\frac{3}{4}$

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

$\Rightarrow P\left(B\right)=\frac{3}{4}+\frac{1}{4}-\frac{1}{3}$ [Using (1) and (2)]

$\therefore P\left(B\right)=\frac{2}{3}$