Problem 36 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

The probability that a woman who resigns works in Store C is 50% or \(\frac{1}{2}\).

Step 1: Understand the given information

We have the following information given: - Store A has 50 employees, with 50% of them being women. - Store B has 75 employees, with 60% of them being women. - Store C has 100 employees, with 70% of them being women. The goal is to find the probability that a woman who resigns works in store C.

Step 2: Calculate the number of women employees in each store

We'll first find the number of women in each store: - Women in Store A: \(50 * 0.50 = 25\) - Women in Store B: \(75 * 0.60 = 45\) - Women in Store C: \(100 * 0.70 = 70\)

Step 3: Calculate the total number of women employees

The next step is to find the total number of women employees among stores A, B, and C: Total Women = Women in A + Women in B + Women in C = 25 + 45 + 70 = 140

Step 4: Calculate the probability of selecting a woman from each store

To find the probability of a woman resigning from Store C, given that a woman resigned, we'll find the probability of selecting a woman from each store: - Probability of selecting a woman from Store A: \(\frac{25}{140}\) - Probability of selecting a woman from Store B: \(\frac{45}{140}\) - Probability of selecting a woman from Store C: \(\frac{70}{140}\)

Step 5: Determine the probability that the woman who resigns works in Store C

Now, we simply need to find the probability that the woman who resigned works in Store C. We have already found the probability for each store: The probability that the woman who resigned works in store C is \(\frac{70}{140}\) or \(\frac{1}{2}\). Hence, the probability that a woman who resigns works in Store C is 50% or \(\frac{1}{2}\).