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.
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.
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Solution and Explanation
Step 1: Assign probabilities
Let: \[ P(E_1) = \frac{4}{7}, \quad P(E_2) = \frac{1}{7}, \quad P(E_3) = \frac{2}{7}. \] The probabilities of increased profits 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
The probability that the profits increased due to \( R \) as CEO is: \[ P(E_3 | A) = \frac{P(E_3) P(A | E_3)}{P(E_1) P(A | E_1) + P(E_2) P(A | E_2) + P(E_3) P(A | E_3)}. \]
Step 3: Substitute the values
\[ P(E_3 | A) = \frac{\frac{2}{7} \cdot 0.5}{\frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5}. \]
Simplify the denominator:
\[ P(E_3 | A) = \frac{\frac{2}{7} \cdot 0.5}{\frac{4}{7} \cdot 0.3 + \frac{1}{7} \cdot 0.8 + \frac{2}{7} \cdot 0.5} = \frac{\frac{1}{7}}{\frac{1.2}{7}} = \frac{1}{3}. \]
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