Question

If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}$ and $P(A\cup B)=\frac{7}{11}$, then the value of $P\left(A/B\right)$ is:

• A. $\frac{2}{3}$
• B. $\frac{4}{5}$
• C. $\frac{4}{11}$
• D. $\frac{5}{6}$

Answer 1

The correct option is B $\frac{4}{5}$

If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}$ and $P(A\cup B)=\frac{7}{11}$,
We know that, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$
$\Rightarrow P(A\cap B)=\frac{6}{11}+\frac{5}{11}-\frac{7}{11}=\frac{11-7}{11}$
$\Rightarrow P(A\cap B)=\frac{4}{11}$
Now by definition of conditional probability,
$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$\Rightarrow P\left(A/B\right)=\frac{4/11}{5/11}$
$\Rightarrow P\left(A/B\right)=\frac{4}{5}$