Question

If $P\left(A\right)=\frac{7}{13}$, $P\left(B\right)=\frac{9}{13}$ ,$P(A\cup B)=\frac{12}{13}$, then find (A|B).

Answer 1

Given:-

$P\left(A\right)=\frac{7}{13}$

$P\left(B\right)=\frac{9}{13}$

$P(A\cup B)=\frac{12}{13}$

As we know that,

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(\cup B)$

$P(A\cap B)=\frac{7}{13}+\frac{9}{13}-\frac{12}{13}$

$\Rightarrow P(A\cap B)=\frac{4}{13}$

Now,

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

$\therefore P\left(A|B\right)=\frac{\left(\frac{4}{13}\right)}{\left(\frac{9}{13}\right)}$

$\Rightarrow P\left(A|B\right)=\frac{4}{9}$

Hence the correct answer is $\frac{4}{9}$.