Question

If $A$ and $B$ are events such that $P\left(A\right)=\frac{1}{3}$, $P\left(B\right)=\frac{1}{4}$ and $P\left(A\cap B\right)=\frac{1}{12}$,then find $P$(not $A$ and not $B$).

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

Answer 1

Answer: (B)

Here $P\left(A\right).P\left(B\right)=\frac{1}{3}\times \frac{1}{4}=\frac{1}{12}=P\left(A\cap B\right)$
$\Rightarrow$ events $A$ and $B$ are independent.
$\Rightarrow$ events $\bar{A}$ and $\bar{B}$ are also independent.
Now $p\left(\bar{A}\cap \bar{B}\right)=P\left(\bar{A}\right)P\left(\bar{B}\right)$
($\because \bar{A}$ and $\bar{B}$ are independent events)
$=\left(1-P\left(A\right)\right)\left(1-P\left(B\right)\right)$
$=\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)=\frac{2}{3}\times \frac{3}{4}$
$=\frac{1}{2}$