Question

If A and B are two events such that P (A) = $\frac{1}{3},$ P (B) = $\frac{1}{5}$ and P (A ∪ B) = $\frac{11}{30}$, find P (A/B) and P (B/A).

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ P\left(A\right)=\frac{1}{3} \\ P\left(B\right)=\frac{1}{5} \\ P\left(A\cup B\right)=\frac{11}{30} \\ \mathrm{Now}, \\ P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow \frac{11}{30}=\frac{1}{3}+\frac{1}{5}-P\left(A\cap B\right) \\ \Rightarrow P\left(A\cap B\right)=\frac{1}{3}+\frac{1}{5}-\frac{11}{30}=\frac{10+6-11}{30}=\frac{5}{30}=\frac{1}{6} \\ P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{1}{5}}}=\frac{5}{6} \\ P\left(B/A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{1}{3}}}=\frac{3}{6}=\frac{1}{2} \end{gathered}$$