Question

The two events $A$ and $B$ have probabilities $0.25$ and $0.50$ respectively.

The probability that both $A$ and $B$ occur simultaneously is $0.14$.

Then the probability that neither $A$ nor $B$ occurs is

• A. $0.39$
• B. $0.25$
• C. $0.904$
• D. None of these

Answer 1

The correct option is A

$0.39$

Explanation for correct option

Given that,

$$\begin{gathered} P\left(A\right)=0.25 \\ P\left(B\right)=0.50 \\ P\left(A\cap B\right)=0.14 \end{gathered}$$

The probability that neither $A$ nor $B$ occurs is $=P\left(A\text{'}\cap B\text{'}\right)$

$$\begin{gathered} =P\left(A\cup B\right)\text{'} \\ =1-P\left(A\cup B\right) \end{gathered}$$

We know that $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

$$\begin{gathered} =0.25+0.50-0.14 \\ =0.61 \end{gathered}$$

$$\begin{gathered} \therefore P\left(A\text{'}\cap B\text{'}\right)=1-P\left(A\cup B\right) \\ =1-0.61 \\ =0.39 \end{gathered}$$

Hence, the correct option is $o\mathrm{ption}\left(\mathrm{A}\right)$