Problem 6 - Basic Statistics for Business and Economics with Student

Question

A firm will promote two employees out of a group of six men and three women. a. List the outcomes of this experiment if there is particular concern about gender equity. b. Which concept of probability would you use to estimate these probabilities?

Step-by-Step Solution

Answer

Outcomes: (2M,0W), (1M,1W), (0M,2W); Use Classical Probability.

Step 1: Define the Context of the Problem

We need to select 2 people out of a total group of 9 people (6 men and 3 women). The concern about gender equity suggests we should look at combinations of genders being selected.

Step 2: List Possible Gender Outcomes

The possible combinations are: (2 men, 0 women), (1 man, 1 woman), (0 men, 2 women). We need to find the number of ways each combination can occur.

Step 3: Calculate Combinations for Each Scenario

For each scenario, we calculate the number of ways to choose the respective number of men and women:1. Two Men, Zero Women: Select 2 men from 6:\[\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15\]2. One Man, One Woman: Select 1 man from 6 and 1 woman from 3:\[\binom{6}{1} \cdot \binom{3}{1} = 6 \cdot 3 = 18\]3. Zero Men, Two Women: Select 2 women from 3:\[\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3\]

Step 4: Determine Total Outcomes

Calculate the total ways to choose any 2 people from a group of 9:\[\binom{9}{2} = \frac{9!}{2!(9-2)!} = 36\]

Step 5: Calculate Probabilities for Each Scenario

Use the combinations from Step 3 over the total from Step 4:1. Probability for Two Men, Zero Women:\[P = \frac{15}{36} = \frac{5}{12}\]2. Probability for One Man, One Woman:\[P = \frac{18}{36} = \frac{1}{2}\]3. Probability for Zero Men, Two Women:\[P = \frac{3}{36} = \frac{1}{12}\]

Step 6: Conclusion on Probabilities

Based on the calculations, the scenario of having one man and one woman has the highest probability. This suggests that combining genders would statistically promote gender equity.

Step 7: Choose the Probability Concept

To estimate these probabilities, we use the "Classical Probability" concept, as there is a finite number of equally likely outcomes in this scenario.

Key Concepts

Combinatorics

In this context, combinatorics is a key mathematical tool used to determine how many different ways we can select individuals from a group. The selection of people without regard to the order in which they are chosen is done through combinations. In our exercise, we are focusing on selecting two employees from a group of six men and three women. Here’s how it breaks down:

• To find the number of ways to select 2 men from 6, we use the combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number, and \( k \) is the number to choose. So, \( \binom{6}{2} = 15 \).
• Similarly, selecting 1 man and 1 woman is done by \( \binom{6}{1} \times \binom{3}{1} = 6 \times 3 = 18 \).
• Choosing 2 women from 3, is calculated by \( \binom{3}{2} = 3 \).

These calculations help in understanding every possible way the two positions can be filled concerning gender distribution.

Gender Equity in Selection

Gender equity in selection refers to ensuring that both men and women have equal opportunities in being chosen for positions or being represented. In our scenario, we want to explore how gender gets distributed when picking two employees for promotion. Through combinatorics, we determine the probability of each gender composition:

• Two men being promoted means that no women are chosen. The probability of this happening is \( \frac{5}{12} \).
• One man and one woman being promoted gives a balanced gender outcome, and its probability is the highest at \( \frac{1}{2} \).
• Zero men and two women being chosen is less likely with a probability of \( \frac{1}{12} \).

These outcomes suggest that the most equitable strategy would naturally tend towards promoting one man and one woman, which aligns with considerations of gender balance.

Classical Probability

Classical probability is utilized here to calculate the likelihood of each gender combination occurring. This method is based on the assumption that all outcomes are equally likely, making it suitable for finite and well-defined sets like our group of employees. To understand it better:

• We first determine the total number of equally possible outcomes, which is selecting any 2 from 9 people: \( \binom{9}{2} = 36 \).
• Using the results from combinatorics, we calculate the probability of each scenario by dividing the number of favorable outcomes by the total number of outcomes.
• The classical probability is straightforward because it relies on simple counting rather than empirical testing or subjective reasoning.

Therefore, with classical probability, we can predict that selecting one man and one woman is the most probable, supporting a balanced selection.