Detailed Solution<\/h3>\r\n <\/div>\r\n\r\n

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Step 1: Define Events and Probabilities

Let $E_1$, $E_2$, and $E_3$ be the events that P, Q, and R are selected as CEO, respectively. Let A be the event that the company increases its profits. We are given the following probabilities:\r\n\r\n$P(E_1) : P(E_2) : P(E_3) = 4 : 1 : 2$\r\n\r\nSo, $P(E_1) = \frac{4}{4+1+2} = \frac{4}{7}$, $P(E_2) = \frac{1}{7}$, and $P(E_3) = \frac{2}{7}$.\r\n\r\nWe are also given the conditional probabilities:\r\n\r\n$P(A|E_1) = 0.3$, $P(A|E_2) = 0.8$, and $P(A|E_3) = 0.5$.

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Step 2: Apply Bayes' Theorem

We want to find the probability that the company's increased profits are due to the appointment of R as CEO, which is $P(E_3|A)$. Using Bayes' Theorem, we have:\r\n\r\n$P(E_3|A) = \frac{P(A|E_3)P(E_3)}{P(A)}$\r\n\r\nWe need to find $P(A)$, which can be calculated using the law of total probability:\r\n\r\n$P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$

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Step 3: Calculate P(A)

Substituting the given values:\r\n\r\n$P(A) = (0.3)(\frac{4}{7}) + (0.8)(\frac{1}{7}) + (0.5)(\frac{2}{7})$\r\n\r\n$P(A) = \frac{1.2}{7} + \frac{0.8}{7} + \frac{1.0}{7} = \frac{1.2 + 0.8 + 1.0}{7} = \frac{3}{7}$

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Step 4: Calculate P(E3|A)

Now, we can find $P(E_3|A)$:\r\n\r\n$P(E_3|A) = \frac{P(A|E_3)P(E_3)}{P(A)} = \frac{(0.5)(\frac{2}{7})}{\frac{3}{7}} = \frac{\frac{1}{7}}{\frac{3}{7}} = \frac{1}{3}$

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\r\n Final Answer: 1/3<\/span>\r\n <\/p>\r\n <\/div>\r\n <\/div>

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