Question

If $A$ and $B$ are two independent events such that $P\left(\bar{A}\right)=0.75$, $P\left(A\cup B\right)=0.65$ and $P\left(B\right)=p$, then find the value of $p$.

• A. $\frac{9}{14}$
• B. $\frac{7}{15}$
• C. $\frac{5}{14}$
• D. $\frac{8}{15}$

Answer 1

Answer: (D)

$P\left(\bar{A}\right)=0.75$
$\Rightarrow P\left(A\right)=1-P\left(\bar{A}\right)=1-0.75=0.25$
Also $A$ and $B$ are independent
$\Rightarrow P\left(A\cap B\right)=P\left(A\right)P\left(B\right)=0.25p$
Using $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$,
$0.65=0.25+p-0.25p$
$\Rightarrow 0.65-0.25=\left(1-0.25\right)p$
$\Rightarrow 0.40=0.75p$
$\Rightarrow p=\frac{0.40}{0.75}=\frac{8}{15}$