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