Question

The board of directors of a major symphony orchestra has voted to create a committee for the purpose of handling employee complaints. The committee will consist of the president and vice president of the symphony board and two orchestra representatives. The two orchestra representatives will be randomly selected from a list of six volunteers, consisting of four men and two women. a. Find the probability distribution for \(x\), the number of women chosen to be orchestra representatives. b. What is the probability that both orchestra representatives will be women? able \(x\).

Short answer

Answer: The probability distribution for the number of women chosen (x) is: - P(x=0) = 2/5 (0 women, 2 men) - P(x=1) = 8/15 (1 woman, 1 man) - P(x=2) = 1/15 (2 women) The probability that both chosen representatives are women is P(x=2) = 1/15.

Step-by-step solution

Calculate the total number of ways to choose two representatives

To find the total number of ways to choose two representatives from the list of six volunteers, we can use the combination formula which is \({n\choose k} = \frac{n!}{k!(n-k)!}\), where n is the total number of elements (in our case, the volunteers) and k is the number of elements to choose (the representatives). In our case, we have n=6 and k=2. Number of ways to choose \(2\) representatives out of \(6\) volunteers: \({6\choose 2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = 15\)

Calculate the probability distribution for \(x\), the number of women chosen

There are three possible scenarios for the number of women chosen: 1. No women are chosen (0 women, 2 men) 2. One woman is chosen (1 woman, 1 man) 3. Both representatives are women (2 women) Now let's calculate the probabilities for each scenario. Scenario 1 (0 women, 2 men): The number of ways to choose 2 men from the 4 men is \({4\choose 2} = 6\) The probability of choosing 0 women and 2 men is \(\frac{6}{15} = \frac{2}{5}\) Scenario 2 (1 woman, 1 man): The number of ways to choose 1 woman from the 2 women is \({2\choose 1} = 2\) The number of ways to choose 1 man from the 4 men is \({4\choose 1} = 4\) The number of ways to choose 1 woman and 1 man is \(2 * 4 = 8\) The probability of choosing 1 woman and 1 man is \(\frac{8}{15}\) Scenario 3 (2 women): The number of ways to choose 2 women from the 2 women is \({2\choose 2} = 1\) The probability of choosing 2 women is \(\frac{1}{15}\) So, the probability distribution for \(x\) is: - P(x=0) = \(\frac{2}{5}\) - P(x=1) = \(\frac{8}{15}\) - P(x=2) = \(\frac{1}{15}\)

Calculate the probability of both representatives being women

From the probability distribution obtained in Step 2, we can directly find the probability of both representatives being women. The probability that both representatives are women is P(x=2) = \(\frac{1}{15}\).

Key concepts

Combination Formula

The combination formula is a powerful tool in probability and combinatorics. It helps us determine the number of ways to select a subset of items from a larger set, without considering the order of selection. The formula is given by:\[{n \choose k} = \frac{n!}{k!(n-k)!}\]where:

• \( n \) is the total number of items,
• \( k \) is the number of items to choose,
• \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \).

In the context of the symphony orchestra's committee, we used the combination formula to determine how many ways we can select 2 members from 6 volunteers. The calculation, \( {6 \choose 2} = 15 \), reveals that there are 15 different ways to form this committee.
This formula is particularly useful when dealing with problems involving random selection, as it allows you to calculate the total number of possible outcomes.

Probability Scenarios

Probability scenarios involve examining different possible outcomes and determining their likelihood. In our committee formation problem, we considered three scenarios:1. **No women are chosen:** - Both representatives are men.2. **One woman is chosen:** - One representative is a woman, and the other is a man.3. **Both representatives are women:** - Both representatives are women.For each scenario, we used combinations to calculate the number of ways to achieve it:

• **Scenario 1:**The probability that 0 women are chosen is calculated by selecting 2 men out of 4 men, \( {4 \choose 2} = 6 \), giving us a probability of \( \frac{2}{5} \).
• **Scenario 2:**The chance of picking one woman and one man is 8 out of the 15, thus \( \frac{8}{15} \).
• **Scenario 3:**Selecting both women gives a probability of \( \frac{1}{15} \).

By dividing the ways to achieve a specific outcome by the total possible outcomes, we can understand how likely each scenario is.

Random Selection

Random selection refers to the process of choosing items where each item has an equal chance of being chosen. This concept is fundamental in probability and statistics when dealing with unbiased experiments.
In the orchestra exercise, random selection is applied to choosing representatives of the committee from volunteers, ensuring fairness. The six volunteers (4 men and 2 women) each have an equal opportunity to be selected.
To analyze random selection:

• **Total Selection Possibilities:** Using the combination formula, we determined there are 15 potential selections of any two volunteers, ensuring each pair is chosen purely by chance.
• **Equitable Probabilities Demonstrated:** Despite the different number of men and women, each scenario—whether selecting no women, one woman, or two women—has been calculated based on equal selection chance.

Understanding random selection helps clarify that every volunteer stood an equal chance of being on the committee, reinforcing that the selection process is unbiased.