The events $A$ and $B$ have probabilities 0.25 and 0.50 , respectively. The probability that both $A$ and…
The 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
$0.39$
$0.29$
$0.11$
$0.25$
Solution
Given that,
$\begin{aligned} P(A) & =0.25, P(B)=0.50 \\ P(A \cap B) & =0.14 \\ P(A \cup B) & =P(A)+P(B)-P(A \cap B) \\ & =0.25+0.50-0.14 \\ & =0.75-0.14=0.61 \\ P(\bar{A} \cap \bar{B}) & =P(\overline{A \cup B}) \\ & =1-P(A \cup B)=1-0.61=0.39\end{aligned}$