The probability of three persons A, B and C becoming clerks of a certain administrative office are 3 ∶ 2 ∶ 4. The probabilities that incentive will be introduced if they become clerks are 0.4, 0.5 and 0.3, respectively. If the incentive has been introduced, then what is the probability that C is appointed as the clerk?

Explanation:

Let the events be described as below:

A: The incentive has been introduced

E₁: Person A appointed as the clerk

E₂: Person B appointed as the clerk

E₃: Person C appointed as the clerk

The probability of three persons A, B, and C becoming clerks of a certain administrative office are 3 ∶ 2 ∶ 4.

Hence, \(P(E_1)=\frac{3}{9},P(E_2)=\frac{2}{9},P(E_3)=\frac{4}{9}\)

The probabilities that incentive will be introduced if they become clerks are 0.4, 0.5, and 0.3, respectively.

Hence, \(P(\frac{A}{E_1})=\frac{4}{10},P(\frac{A}{E_2})=\frac{5}{10},P(\frac{A}{E_3})=\frac{3}{10} \)

∴ \(P(A)=P(E_1).P(\frac{A}{E_1})+P(E_2).P(\frac{A}{E_2})+P(E_3).P(\frac{A}{E_3}) \)

\(P(A)=\frac{3}{9}\times \frac{4}{10}+\frac{2}{9}\times \frac{5}{10} +\frac{4}{9}\times \frac{3}{10} \)

\(P(A)=\frac{12}{90}+\frac{10}{90}+\frac{12}{90}=\frac{34}{90}\)

∴ the required probability is

\(P(\frac{E_3}{A})=\frac{P(\frac{A}{E_3}).P(E_3)}{P(A)}\)

\(=\frac{\frac{3}{10}.\frac{4}{9}}{\frac{34}{90}}\)

\(=\frac{12}{90}\times \frac{90}{34}\)

\(=\frac{6}{17}\)