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 \(\frac{1}{2}\).

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in statistics that helps us determine the probability of an event occurring, given that another event has already occurred. In this exercise, we are interested in calculating the probability that a woman who resigns from one of the stores works at Store C. This involves using conditional probability because we know the resigning employee is a woman, which affects the probability calculation.

To apply conditional probability, we first find the total number of women employees across all stores and then the number in each individual store. The probability of the resigned woman being from Store C is calculated by dividing the number of women in Store C by the total number of women across the stores. This process ensures we only consider women employees, which is the given condition for this probability calculation.

Bayesian Inference

Bayesian inference is a method of statistical inference in which Bayes' Theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Although the exercise doesn't explicitly involve Bayesian inference, understanding it provides insight into similar probabilistic reasoning processes.

Bayes' Theorem helps us revise our expectations (probabilities) by taking new data into account. In simple terms, if we consider every piece of data or evidence, it allows us to calculate the likelihood of future events. In this exercise, we could use a similar thought process to update our belief about which store a resigning woman might work at if we were given supplementary information beyond the initial conditions about employees and their genders.

Statistics

Statistics is the discipline that deals with collecting, analyzing, interpreting, and presenting data. This exercise is a good demonstration of how statistics can be used to make informed decisions based on available data. By calculating probabilities, we are essentially summarizing complex data into an understandable form.

In statistics, understanding descriptive measures like counts and percentages can help in preliminary data analysis. Here, we use simple arithmetic to find the number of women employees in each store and the total number of women employees. These basic statistical measures are foundational for calculating the probabilities of interest and showcasing the real-world utility of statistical methods in decision-making scenarios.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of elements in sets. While our exercise doesn't dive deep into complex combinatorial problems, it subtly relies on these principles for probability calculations.

When determining the number of women in each store and subsequently calculating probabilities, we're essentially engaging in a simple combinatorial process. We count and combine the possible outcomes of resignations (women employees) to calculate the desired probability. Such combinatorial reasoning underpins many probability problems and provides a clear, logical way to structure complex calculations.