Question

If A and B are two independent events such that P ($\overline{A}$ ∩ B) = 2/15 and P (A ∩ $\overline{B}$) = 1/6, then find P (B).

Answer 1

$$\begin{gathered} \mathrm{Let}: \\ P\left(A\right)=x \\ P\left(B\right)=y \\ P\left(\overline{A}\cap B\right)=\frac{2}{15} \\ \Rightarrow P\left(\overline{A}\right)\times P\left(B\right)=\frac{2}{15} \\ \Rightarrow \left(1-x\right)y=\frac{2}{15}...\left(1\right) \\ P\left(A\cap \overline{B}\right)=\frac{1}{6} \\ \Rightarrow P\left(A\right)\times P\left(\overline{B}\right)=\frac{1}{6} \\ \Rightarrow \left(1-y\right)x=\frac{1}{6}...\left(2\right) \\ \mathrm{Subtracting}\left(2\right)\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get} \\ x-y=\frac{1}{30} \\ x=y+\frac{1}{30} \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{x}\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ \left(y+\frac{1}{30}\right)\left(1-y\right)=\frac{1}{6} \\ \Rightarrow 30y^2-29y+4=0 \\ \Rightarrow y=\frac{1}{6},\frac{4}{5} \end{gathered}$$