Question

If A and B are two events such that
(i) P (A) = 1/3, P (B) = 1/4 and P (A ∪ B) = 5/12, find P (A/B) and P (B/A).
(ii) P (A) = $\frac{6}{11},$ P (B) = $\frac{5}{11}$ and P (A ∪ B) = $\frac{7}{11},$ find P (A ∩ B), P (A/B), P (B/A).
(iii) P (A) = $\frac{7}{13},$ P (B) = $\frac{9}{13}$ and P (A ∩ B) = $\frac{4}{13},$ find $P\left(\overline{A}/B\right).$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Given}: \\ P\left(A\right)=\frac{1}{3} \\ P\left(B\right)=\frac{1}{4} \\ P\left(A\cup B\right)=\frac{5}{12} \\ P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow \frac{5}{12}=\frac{1}{3}+\frac{1}{4}-P\left(A\cap B\right) \\ \Rightarrow P\left(A\cap B\right)=\frac{1}{3}+\frac{1}{4}-\frac{5}{12}=\frac{2}{12}=\frac{1}{6} \\ \mathrm{Now}, \\ P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{1}{4}}}=\frac{4}{6}=\frac{2}{3} \\ P\left(B/A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{1}{3}}}=\frac{3}{6}=\frac{1}{2} \end{gathered}$$

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

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Given}: \\ P\left(A\right)=\frac{7}{13} \\ P\left(B\right)=\frac{9}{13} \\ P\left(A\cap B\right)=\frac{4}{13} \\ \mathrm{Now}, \\ P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{{\displaystyle \frac{4}{13}}}{{\displaystyle \frac{9}{13}}}=\frac{4}{9} \\ P\left(\overline{A}/B\right)=1-P\left(A/B\right)=1-\frac{4}{9}=\frac{5}{9} \end{gathered}$$