Question

Stores \(A, B\), and \(C\) have 50,75, and 100 employees, and, respectively, 50,60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in-store \(C\)?

Short answer

The probability that the woman who resigns works in store C is approximately 0.426 or 42.6%.

Step-by-step solution

Finding the number of women in each store

We know the total number of employees and the percentage of women in each store. We can find the number of women in each store using the provided percentages. For store A: \(50 \cdot 0.5 = 25\) women For store B: \(75 \cdot 0.6 = 45\) women For store C: \(100 \cdot 0.7 = 70\) women

Finding the probability of a woman resigning from each store

Since resignations are equally likely among all employees, the probability of a woman resigning from a store is the number of women in that store divided by the total number of employees: For store A: \(P(W_A) = \frac{25}{50} = 0.5\) For store B: \(P(W_B) = \frac{45}{75} = 0.6\) For store C: \(P(W_C) = \frac{70}{100} = 0.7\)

Finding the total probability of a woman resigning

The total probability of a woman resigning is the sum of the probabilities of a woman resigning from each store, weighted by the number of employees in each store. \(P(W) = P(W_A) \cdot \frac{50}{225} + P(W_B) \cdot \frac{75}{225} + P(W_C) \cdot \frac{100}{225}\) Plugging in the values we found in Step 2: \(P(W) = 0.5 \cdot \frac{50}{225} + 0.6 \cdot \frac{75}{225} + 0.7 \cdot \frac{100}{225}\) \(P(W) = 0.2222 + 0.2 + 0.3111 = 0.7333\)

Applying Bayes' theorem

We want to find the probability that a woman who resigns works in store C. We can use the Bayes' theorem: \(P(C|W) = \frac{P(W|C) \cdot P(C)}{P(W)}\) Where, - \(P(C|W)\) is the probability we want to find (woman resigning was from store C), - \(P(W|C)\) is the probability of a woman resigning from store C (found in Step 2), - \(P(C)\) is the probability of randomly selecting store C (based on the number of employees), - \(P(W)\) is the total probability of a woman resigning (found in Step 3).

Calculating the probability of randomly selecting store C

We calculate the probability of randomly selecting store C using the total number of employees: \(P(C) = \frac{\text{Number of employees in store C}}{\text{Total number of employees}}\) \(P(C) = \frac{100}{225}\)

Putting it all together

Now we have all the values we need to plug into the Bayes' theorem formula: \(P(C|W) = \frac{P(W|C) \cdot P(C)}{P(W)}\) \(P(C|W) = \frac{0.7 \cdot \frac{100}{225}}{0.7333}\) \(P(C|W) \approx 0.426\) Therefore, the probability that the woman who resigns works in store C is approximately 0.426 or 42.6%.