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. Define events: Let \( P_1, P_2, P_3 \) represent the selections of \( P, Q, \) and \( R \) as CEO, respectively. Let \( E \) denote the event that the company increases profits.
2. Apply Bayes' theorem: The probability to be calculated is \( P(P_3 \,|\, E) = \frac{P(P_3) \cdot P(E \,|\, P_3)}{P(E)}. \)
3. Determine prior probabilities: Based on the ratio \( 4 : 1 : 2 \), the prior probabilities are \( P(P_1) = \frac{4}{7}, \quad P(P_2) = \frac{1}{7}, \quad P(P_3) = \frac{2}{7}. \)
4. Calculate total probability \( P(E) \): Using the law of total probability, \( P(E) = P(P_1) \cdot P(E \,|\, P_1) + P(P_2) \cdot P(E \,|\, P_2) + P(P_3) \cdot P(E \,|\, P_3) \). Substituting the given probabilities: \( P(E) = \frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5 \). Simplifying yields \( P(E) = \frac{1.2}{7} + \frac{0.8}{7} + \frac{1.0}{7} = \frac{3.0}{7} \).
5. Compute conditional probability \( P(P_3 \,|\, E) \): Using Bayes' theorem, \( P(P_3 \,|\, E) = \frac{P(P_3) \cdot P(E \,|\, P_3)}{P(E)} \). Substituting values: \( P(P_3 \,|\, E) = \frac{\frac{2}{7} \cdot 0.5}{\frac{3.0}{7}} = \frac{1.0}{3.0} = \frac{1}{3} \).
Final Answer: The probability that the increase in profits is attributable to \( R \)'s appointment as CEO is \( \boxed{\frac{1}{3}} \).