Question

Evaluate $P(A\cup B)$, if $2P\left(A\right)=P\left(B\right)=\frac{5}{13}$ and $P\left(A|B\right)=\frac{2}{5}$. Find $P(A\cup B)\times 100$

Answer 1

It is given that, $2P\left(A\right)=P\left(B\right)=\frac{5}{13}$

$\Rightarrow P\left(A\right)=\frac{5}{26}$ and $P\left(B\right)=\frac{5}{13}$

$P\left(A|B\right)=\frac{2}{5}$

$\Rightarrow \frac{P(A\cap B)}{P\left(B\right)}=\frac{2}{5}$

$\Rightarrow P(A\cap B)=\frac{2}{5}\times P\left(B\right)=\frac{2}{5}\times \frac{5}{13}=\frac{2}{13}$

It is known that, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow P(A\cup B)=\frac{5}{26}+\frac{5}{13}-\frac{2}{13}$

$\Rightarrow P(A\cup B)=\frac{5+10-4}{26}$

$\Rightarrow P(A\cup B)=\frac{11}{26}=0.42$

$\Rightarrow P(A\cup B)\times 100=42$