Question

A and B are two independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of occurrence of two events.

Answer 1

$$\begin{gathered} P\left(A\cap B\right)=P\left(A\right)P\left(B\right)\left(\text{A and B are independent even}\mathrm{ts}\right) \\ \frac{1}{6}=P\left(A\right)P\left(B\right) \\ \Rightarrow P\left(A\right)=\frac{1}{6P\left(B\right)}...\left(1\right) \\ P\left(\overline{A}\cap \overline{B}\right)=\left[1-P\left(A\right)\right]\left[1-P\left(B\right)\right] \\ \Rightarrow \frac{1}{3}=\left[1-P\left(A\right)\right]\left[1-P\left(B\right)\right] \\ \Rightarrow \frac{1}{3}=\left[1-\frac{1}{6P\left(B\right)}\right]\left[1-P\left(B\right)\right]\left[\mathrm{Using}\left(1\right)\right] \\ \text{Let}P\left(B\right)=x \\ \Rightarrow \left(\frac{6x-1}{6x}\right)\left(1-x\right)=\frac{1}{3} \\ \Rightarrow 6x-1-6x^2+x=2x \\ \Rightarrow 6x^2-5x+1=0 \\ \Rightarrow \left(2x-1\right)\left(3x-1\right)=0 \\ \Rightarrow x=\frac{1}{2}\mathrm{or}x=\frac{1}{3} \\ \mathrm{If}P\left(B\right)=\frac{1}{2},\text{then}P\left(A\right)=\frac{1}{3} \\ \mathrm{If}P\left(B\right)=\frac{1}{3},\text{then}P\left(A\right)=\frac{1}{2} \end{gathered}$$