Question

Job discrimination? A company with a large sales staff announces openings for three positions as regional managers. Twenty-two of the current salespersons apply, 12 men and 10 women. After the interviews, when the company announces the newly appointed managers, all three positions go to women. The men complain of job discrimination. Do they have a case? Simulate a random selection of three people from the applicant pool, and make a decision about the likelihood that a fair process would result in hiring all women.

Short answer

The probability of randomly selecting 3 women is about 7.79%, suggesting potential job discrimination.

Step-by-step solution

Understand the Process

We need to determine if the selection of 3 women out of 3 positions from a pool of 12 men and 10 women was likely to happen by chance in a fair selection process.

Set up the Probability Model

Use a hypergeometric distribution to model the situation, where we are drawing 3 women from a group of 22 applicants (12 men and 10 women).

Calculate the Probability

The probability of choosing 3 women (with no men being selected) from 22 applicants is calculated using the formula for hypergeometric distribution: \[ P(X = 3) = \frac{{\binom{10}{3} \cdot \binom{12}{0}}}{{\binom{22}{3}}} \]

Evaluate Combinatorial Terms

Calculate each of the binomial coefficients: \[ \binom{10}{3} = 120, \quad \binom{12}{0} = 1, \quad \binom{22}{3} = 1540 \]

Compute the Final Probability

Plug the values into the formula to find the probability: \[ P(X = 3) = \frac{120}{1540} \approx 0.0779 \] This is approximately 7.79%.

Interpret the Probability

The probability of a fair selection resulting in only women being chosen is about 7.79%, which is relatively unlikely in a fair process.

Conclusion

Given that the probability is low, the men's claim of job discrimination may be supported by the fact that such an event is unlikely if all candidates were equally likely to be chosen.

Key concepts

Hypergeometric Distribution

The hypergeometric distribution is a probability distribution that describes the likelihood of drawing a specific number of successes (items of interest) from a finite population without replacement.
In the context of the job discrimination problem, it helps model the probability of selecting a certain number of women from the applicant pool when hiring for three positions.
To understand this, consider the characteristics of a hypergeometric distribution:

• It deals with two types of outcomes (e.g., women and men here).
• The population size is fixed (22 applicants in total).
• There is no replacement, meaning each position filled affects the probabilities of the remaining selections.

The probability mass function of a hypergeometric distribution calculates the likelihood of obtaining exactly \( k \) successes (women selected) in \( n \) draws (3 jobs), given \( N \) successes in the population (10 women) and \( K \) draws remaining (22 total applicants). The formula used is:\[P(X = k) = \frac{{\binom{N}{k} \cdot \binom{M - N}{n-k}}}{{\binom{M}{n}}}\]Where:- \( N = 10 \) women,- \( M = 22 \) total candidates,- \( n = 3 \) positions.- \( k = 3 \) women selected.This allows us to compute probabilities and determine the likelihood of such hiring outcomes.

Combinatorial Mathematics

Combinatorial mathematics is a branch of mathematics dealing with the study of countable, discrete structures. It is the art of counting, arranging, and optimizing arrangements of different elements. In our problem, combinatorial mathematics helps calculate the number of ways to choose committee members.
In a hypergeometric distribution problem, we often use binomial coefficients to count combinations, which are central to solving such problems.

• A binomial coefficient \( \binom{n}{k} \) represents the number of ways to select \( k \) elements from a set of \( n \) elements.
• It's calculated as \( \frac{n!}{k!(n-k)!} \), where \(! \) denotes factorial, the product of all positive integers up to a given number.

In the exercise, we calculate:

• \( \binom{10}{3} = 120 \) ways to choose 3 women from 10.
• \( \binom{12}{0} = 1 \) way to choose 0 men from 12 (since there's only one way not to choose any).
• \( \binom{22}{3} = 1540 \) total ways to choose 3 applicants from 22, irrespective of gender.

These calculations help us understand the possible combinations and probabilities involved, furthering our analysis of the situation.

Gender Discrimination in Hiring

Gender discrimination in hiring refers to the unfair treatment of individuals based on their gender, affecting employment decisions. This notion is significant in situations where outcomes, like in the company scenario, diverge considerably from expectations if a fair process was applied.
In our given problem, the probability of hiring three women out of ten applicants, when analyzed using hypergeometric distribution, was determined to be a mere 7.79%. This low probability suggests that randomly and fairly selecting all women for the positions is an unlikely event, hinting at possible gender discrimination in the hiring process.
How to detect possible discrimination?

• Analyzing the statistical likelihood using probability models like the hypergeometric distribution.
• Comparing observed outcomes with expected outcomes under fair conditions.
• Gathering qualitative evidence such as testimonies or policies that show bias against a particular gender.

It is essential to consider these statistical analyses alongside other evidence when investigating hiring discrimination. The numerical probability of unlikely outcomes can provide a significant initial indication but should always be validated with comprehensive assessments.