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.
Step 1: Determine initial probabilities for each CEO and the probability of increased profits given each CEO.
Let the probabilities of each CEO being in charge be: \( P(E_1) = \frac{4}{7}, \quad P(E_2) = \frac{1}{7}, \quad P(E_3) = \frac{2}{7}. \) The probabilities of increased profits (Event A) under each CEO are: \( P(A | E_1) = 0.3, \quad P(A | E_2) = 0.8, \quad P(A | E_3) = 0.5. \)
Step 2: Apply Bayes' theorem to find the posterior probability of a specific CEO being in charge given increased profits.
The probability that the profits increased due to \( E_3 \) being the CEO is calculated as: \[ P(E_3 | A) = \frac{P(E_3) P(A | E_3)}{P(E_1) P(A | E_1) + P(E_2) P(A | E_2) + P(E_3) P(A | E_3)}. \]
Step 3: Substitute the known probabilities into the formula and compute the result.
\[ P(E_3 | A) = \frac{\frac{2}{7} \cdot 0.5}{\frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5}. \]
Calculate the numerator and the denominator:
\[ P(E_3 | A) = \frac{\frac{2}{7} \cdot 0.5}{\frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5} = \frac{\frac{1}{7}}{\frac{1.2}{7}} = \frac{1}{3}. \]