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: - \( P_1, P_2, P_3 \): \( P, Q, \) and \( R \) are selected as CEO, respectively. - \( E \): The company experiences an increase in 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: Given the 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. Compute Total Probability of Event \( E \): \[P(E) = P(P_1) \cdot P(E \,|\, P_1) + P(P_2) \cdot P(E \,|\, P_2) + P(P_3) \cdot P(E \,|\, P_3).\] Substitute the given values: \[P(E) = \frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5.\] 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 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 Result: The probability that the profit increase is a result of \( R \)'s CEO appointment is \( \boxed{\frac{1}{3}} \).