Question

If P (A) = $\frac{6}{11},$ P (B) = $\frac{5}{11}$ and P (A ∪ B) = $\frac{7}{11},$ find
(i) P (A ∩ B)
(ii) P (A/B)
(iii) P (B/A)

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ P\left(A\right)=\frac{6}{11} \\ P\left(B\right)=\frac{5}{11} \\ P\left(A\cup B\right)=\frac{7}{11} \\ \left(\mathrm{i}\right)P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow \frac{7}{11}=\frac{6}{11}+\frac{5}{11}-P\left(A\cap B\right) \\ \Rightarrow P\left(A\cap B\right)=\frac{6}{11}+\frac{5}{11}-\frac{7}{11}=\frac{4}{11} \\ \left(\mathrm{ii}\right)P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)} \\ =\frac{{\displaystyle \frac{4}{11}}}{{\displaystyle \frac{5}{11}}} \\ =\frac{4}{5} \\ \left(\mathrm{iii}\right)P\left(B/A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)} \\ =\frac{{\displaystyle \frac{4}{11}}}{{\displaystyle \frac{6}{11}}} \\ =\frac{4}{6} \\ =\frac{2}{3} \end{gathered}$$