Question

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

• A. $0.6$
• B. $0.2$
• C. $0.21$
• D. None of these

Answer 1

The correct option is B

$0.2$

Calculate the probability

Suppose, probability of an event $A$ occurring is $P\left(A\right)$ and that of $B$is $P\left(B\right)$

$\therefore P\left(A\right)=0.5$ and $P\left(B\right)=0.3$

Now we need to find the probability of neither $A$ nor$B$, which is $P\left(A^{\text{'}}\cap B^{\text{'}}\right)$

$$\begin{gathered} P\left(A^{\text{'}}\cap B^{\text{'}}\right)=P\left(\overline{A\cup B}\right) \\ \Rightarrow P\left(A^{\text{'}}\cap B^{\text{'}}\right)=1-P\left(A\cup B\right) \\ \Rightarrow P\left(A^{\text{'}}\cap B^{\text{'}}\right)=1-[P\left(A\right)+P\left(B\right)] \end{gathered}$$

Now the events are mutually exclusive, so $P\left(A\cap B\right)=0$

$$\begin{gathered} P\left(A^{\text{'}}\cap B^{\text{'}}\right)=1-\left[0.5+0.3\right] \\ \Rightarrow P\left(A^{\text{'}}\cap B^{\text{'}}\right)=1-0.8 \\ \Rightarrow P\left(A^{\text{'}}\cap B^{\text{'}}\right)=0.2 \end{gathered}$$

Therefore, the probability of neither $A$ nor $B$ is $0.2$, so, option (B) is the correct answer.