Question

The chances of \( P \), \( Q \), and \( R \) getting selected as CEO of a company are in the ratio \( 4 : 1 : 2 \), respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, \( P, Q, \) or \( R \), are \( 0.3, 0.8, \) and \( 0.5 \), respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of \( R \) as CEO.

Hint:

Use Bayes' theorem for conditional probabilities: \( P(A \,|\, B) = \frac{P(A) \cdot P(B \,|\, A)}{P(B)} \), and calculate \( P(B) \) using the law of total probability.

Answer 1

1. Let the events be: - \( P_1, P_2, P_3 \): Selection of \( P, Q, \) and \( R \) as CEO. - \( E \): Company increases profits. 
2. Use Bayes' theorem: The required probability is: \[ P(P_3 \,|\, E) = \frac{P(P_3) \cdot P(E \,|\, P_3)}{P(E)}. \] 
3. Calculate the prior probabilities: From the given ratio \( 4 : 1 : 2 \): \[ P(P_1) = \frac{4}{7}, \quad P(P_2) = \frac{1}{7}, \quad P(P_3) = \frac{2}{7}. \] 
4. Calculate the total probability \( P(E) \): \[ P(E) = P(P_1) \cdot P(E \,|\, P_1) + P(P_2) \cdot P(E \,|\, P_2) + P(P_3) \cdot P(E \,|\, P_3). \] Substitute the given probabilities: \[ P(E) = \frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5. \] Simplify: \[ P(E) = \frac{1.2}{7} + \frac{0.8}{7} + \frac{1.0}{7} = \frac{3.0}{7}. \] 
5. Calculate \( P(P_3 \,|\, E) \): \[ P(P_3 \,|\, E) = \frac{P(P_3) \cdot P(E \,|\, P_3)}{P(E)}. \] Substitute: \[ P(P_3 \,|\, E) = \frac{\frac{2}{7} \cdot 0.5}{\frac{3.0}{7}} = \frac{1.0}{3.0} = \frac{1}{3}. \] 
Final Answer: The probability that the increase in profits is due to \( R \)'s appointment as CEO is \( \boxed{\frac{1}{3}} \).