Question

If A and B are two events such that 2 P (A) = P (B) = $\frac{5}{13}$ and P (A/B) = $\frac{2}{5},$ find P (A ∪ B).

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ 2P\left(A\right)=P\left(B\right)=\frac{5}{13} \\ P\left(A/B\right)=\frac{2}{5} \\ \therefore P\left(A\right)=\frac{5}{26} \\ P\left(B\right)=\frac{5}{13} \\ \mathrm{Now}, \\ P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)} \\ \Rightarrow \frac{2}{5}=\frac{P\left(A\cap B\right)}{{\displaystyle \frac{5}{13}}} \\ \Rightarrow P\left(A\cap B\right)=\frac{2}{5}\times \frac{5}{13}=\frac{2}{13} \\ P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ =\frac{5}{26}+\frac{5}{13}-\frac{2}{13} \\ =\frac{11}{26} \end{gathered}$$