Question

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

Answer 1

To find : $P(A\cup B)$

Given: $2P\left(A\right)=P\left(B\right)=\frac{5}{13},P\left(A|B\right)=\frac{2}{5}$

$\Rightarrow P\left(A\right)=\frac{5}{2\times 13}=\frac{5}{26}$

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

We know that $P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}$

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

We know that $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{5}{26}+\frac{5}{13}-\frac{2}{13}=\frac{5+10-4}{26}=\frac{11}{26}$