CBSE 2024, Set 3

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.

Edvaya · Accepted Answer

The correct answer is Subjective. **Step 1: Define events and their initial probabilities**

Let $E_1$, $E_2$, and $E_3$ be the events of selecting P, Q, and R as CEO, respectively.
Let $A$ be the event that the company increases its profits.

From the given ratio $4:1:2$, the sum of the ratio terms is $7$:
$P(E_1) = \frac{4}{7}$
$P(E_2) = \frac{1}{7}$
$P(E_3) = \frac{2}{7}$

**Step 2: Define conditional probabilities**

The probabilities of increasing profit given a specific CEO are:
$P(A|E_1) = 0.3 = \frac{3}{10}$
$P(A|E_2) = 0.8 = \frac{8}{10}$
$P(A|E_3) = 0.5 = \frac{5}{10}$

**Step 3: Apply Bayes' Theorem**

We need to find the probability that R was appointed given that profits increased, which is $P(E_3|A)$.

Using Bayes' Theorem:
$$P(E_3|A) = \frac{P(E_3) \cdot P(A|E_3)}{P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2) + P(E_3) \cdot P(A|E_3)}$$

Calculate the Numerator:
$$= \left(\frac{2}{7}ight)\left(\frac{5}{10}ight) = \frac{10}{70}$$

Calculate the Denominator (Total Probability):
$$= \left(\frac{4}{7}ight)\left(\frac{3}{10}ight) + \left(\frac{1}{7}ight)\left(\frac{8}{10}ight) + \left(\frac{2}{7}ight)\left(\frac{5}{10}ight)$$
$$= \frac{12}{70} + \frac{8}{70} + \frac{10}{70}$$
$$= \frac{30}{70}$$

**Step 4: Final Calculation**

$$P(E_3|A) = \frac{\frac{10}{70}}{\frac{30}{70}} = \frac{10}{30} = \frac{1}{3}$$

**Therefore, the required probability is $\frac{1}{3}$.**

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