Question

If P (not B) = 0.65, P (A ∪ B) = 0.85, and A and B are independent events, then find P (A).

Answer 1

$$\begin{gathered} P\left(\overline{B}\right)=0.65 \\ \Rightarrow 1-P\left(B\right)=0.65 \\ \Rightarrow P\left(B\right)=1-0.65=0.35 \\ \mathrm{Now}, \\ P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\right)\times P\left(B\right) \\ \Rightarrow 0.85=P\left(A\right)+0.35-0.35\times P\left(A\right) \\ \Rightarrow 0.85-0.35=P\left(A\right)\left[1-0.35\right] \\ \Rightarrow P\left(A\right)=\frac{0.5}{0.65}=0.77 \end{gathered}$$