Question

The probability of happening of an event $A$ is $0.5$ and that of $B$ is $0.3$. If $A$ and $B$ are mutually exclusive events, then the probability of neither $A$ nor $B$ is

Answer 1

It is given that
$P\left(A\right)=0.5P\left(B\right)=0.3$

Also , $AandB$ are mutually exclusive events.
$\therefore P(A\cap B)=0$
$\text{To find the value of P neither A nor B}$

$=P(\overline{A}\cap \overline{B})$

Now, $P(\overline{A}\cap \overline{B})=P\overline{(A\cap B)}$

$$\left[\begin{array}{c}ApplyingDe-Morga{n}^{\prime }sLaw\\ P(\overline{X}\cap \overline{Y})=P\left(\overline{X\cap Y}\right)\end{array}\right]$$

= $1-P(A\cup B)$

$$\left[\begin{array}{c}Applyingcompletmentrule\\ P\left(X\right)+P\left(\overline{X}\right)=1\end{array}\right]$$

$P(\overline{A}\cap \overline{B})=1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]$

[Appyling addition rule of probability ]

$=1-(0.5+0.3-0)$
$=1-0.8$
$=0.2$