Question

The value of $P(A\cup B)$
if $2P\left(A\right)=P\left(B\right)=\frac{5}{13}$ and $P\left(A/B\right)=\frac{2}{5}$ is:

• A. $\frac{11}{23}$
• B. $\frac{11}{26}$
• C. $\frac{1}{2}$
• D. $\frac{15}{26}$

Answer 1

The correct option is B $\frac{11}{26}$

Given $2P\left(A\right)=P\left(B\right)=\frac{5}{13}$ and $P\left(A/B\right)=\frac{2}{5}$
$\Rightarrow P\left(B\right)=\frac{5}{13},P\left(A\right)=\frac{5}{13\times 2}=\frac{5}{26}\cdots \left(i\right)$
We know that by definition of conditional probability,
$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$\Rightarrow P(A\cap B)=P\left(A/B\right)P\left(B\right)$
$\Rightarrow P(A\cap B)=\frac{2}{5}\times \frac{5}{13}=\frac{2}{13}\cdots \left(ii\right)$
Now, $\therefore P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow P(A\cup B)=\frac{5}{26}+\frac{5}{13}-\frac{2}{13}=\frac{5+10-4}{26}$
$\Rightarrow P(A\cup B)=\frac{11}{26}$