Question

If A and B are two independent events such that P (A) = 0.3 and P (A ∪ $\overline{B}$) = 0.8. Find P (B).

Answer 1

$$\begin{gathered} \text{A and B are two independent events.} \\ \therefore P\left(A\cup \overline{B}\right)=P\left(A\right)+P\left(\overline{B}\right)-P\left(A\cap \overline{B}\right) \\ \Rightarrow 0.8=0.3+\left[1-P\left(B\right)\right]-P\left(A\right)P\left(\overline{B}\right) \\ \Rightarrow 0.5=1-P\left(B\right)-0.3\left[1-P\left(B\right)\right] \\ \Rightarrow 0.5=1-P\left(B\right)-0.3+O.3P\left(B\right) \\ \Rightarrow 0.5=0.7-P\left(B\right)\left[1-0.3\right] \\ \Rightarrow 0.7P\left(B\right)=0.2 \\ \Rightarrow P\left(B\right)=\frac{0.2}{0.7}=\frac{2}{7} \end{gathered}$$