Question

For two event A and B, if P(A) = P(A/B) = $\frac{1}{4}\mathrm{and}P(B/A)=\frac{1}{2}$, then A and B are _____________ events.

Answer 1

For two events A and B
P(A) = P(A/B) = $\frac{1}{4}\mathrm{and}P(B/A)=\frac{1}{2}$

$$\begin{gathered} \mathrm{i}.\mathrm{e}\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{1}{4}\mathrm{and}\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{1}{2} \\ \mathrm{i}.\mathrm{e}P\left(A\cap B\right)=\frac{1}{4}P\left(B\right) \\ \mathrm{i}.\mathrm{e}P\left(A\cap B\right)=P\left(A\right)P\left(B\right)\left[\because P\left(A\right)=\frac{1}{4}\mathrm{given}\right] \end{gathered}$$
∴ A and B are independent events.