Question

Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.

Answer 1

Let A, E₁ and E₂ denote the events that the person is a good orator, is a man and is a woman, respectively.

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{1}{2} \\ P\left(E_2\right)=\frac{1}{2} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=\frac{5}{100} \\ P\left(A/E_2\right)=\frac{25}{1000} \\ \mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)} \\ =\frac{{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{5}{100}}}{{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{5}{100}}+{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{25}{1000}}} \\ =\frac{{\displaystyle 1}}{1+{\displaystyle \frac{1}{2}}}=\frac{2}{3} \end{gathered}$$