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Chapter 4: Problem 89

A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let \(x\) equal the number of women chosen to fill the two positions. a. Find \(p(x)\). b. Construct a probability histogram for \(x\).

Short Answer

Expert verified
Answer: The probability distribution of the number of women chosen is as follows: - For x = 0 (no woman chosen), the probability p(x=0) = 3/10. - For x = 1 (one woman chosen), the probability p(x=1) = 6/10. - For x = 2 (two women chosen), the probability p(x=2) = 1/10. The probability histogram can be represented with three bars for each value of x: - A bar with a height of 3/10 for x = 0. - A bar with a height of 6/10 for x = 1. - A bar with a height of 1/10 for x = 2.

Step by step solution

01

In this problem, there are five applicants with two women and three men. We have to fill 2 positions out of these 5 applicants. The possible values for \(x\), the number of women chosen, are 0, 1, and 2. #Step 2: Calculate the total number of ways to choose two applicants#

We can use the formula for combinations to calculate the total number of ways we can choose 2 applicants out of 5: \(\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10\). #Step 3: Calculate the probabilities p(x) for each value of x#
02

x=0: No woman is chosen

In this case, both positions are filled with men. The number of ways to choose 2 men from the 3 available is: \(\binom{3}{2} = 3\). So, the probability \(p(x=0) = \frac{\text{number of ways to choose 2 men}}{\text{total number of ways to choose 2 applicants}} = \frac{3}{10}\).
03

x=1: One woman is chosen

In this case, we are choosing one woman and one man. The number of ways to choose 1 woman from the 2 available is: \(\binom{2}{1} = 2\). And, the number of ways to choose 1 man from the 3 available is: \(\binom{3}{1} = 3\). Therefore, the total number of ways to choose 1 woman and 1 man is \(2 \times 3 = 6\). The probability \(p(x=1) = \frac{\text{number of ways to choose 1 woman and 1 man}}{\text{total number of ways to choose 2 applicants}} = \frac{6}{10}\).
04

x=2: Two women are chosen

In this case, both positions are filled with women. The number of ways to choose 2 women from the 2 available is: \(\binom{2}{2} = 1\). So, the probability \(p(x=2) = \frac{\text{number of ways to choose 2 women}}{\text{total number of ways to choose 2 applicants}} = \frac{1}{10}\). #b. Construct a probability histogram for \(x\)#
05

Plotting the probability histogram

To construct the probability histogram, we will use the values of \(x\) and their corresponding probabilities \(p(x)\) calculated in the previous steps. The histogram should have three bars representing the probabilities of 0, 1, or 2 women being chosen for the two positions. Here is a representation of the probability histogram: x-values: 0 1 2 Probabilities: 3/10 6/10 1/10 So, the probability histogram will have a bar with height 3/10 for x=0, a bar with height 6/10 for x=1, and a bar with height 1/10 for x=2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combination Formula
The combination formula is a key concept in probability and statistics used to determine the number of ways to choose a subset of items from a larger set without considering the order. In the context of our exercise, we need to select 2 applicants out of a total of 5. The combination formula is represented mathematically as:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
where:
  • \( n \) is the total number of items to choose from,
  • \( r \) is the number of items to choose,
  • \( ! \) denotes the factorial operation, which multiplies all whole numbers from the chosen number down to 1.
In our case, we have \( n = 5 \) and \( r = 2 \), so the total number of combinations is:
\[\binom{5}{2} = \frac{5!}{2! \times 3!} = 10\]
This means there are 10 possible ways to select 2 applicants from the 5.
Probability Distribution
A probability distribution represents the likelihood of each possible outcome in an experiment or scenario. It provides a map of all the potential values a random variable can take and their corresponding probabilities. In our exercise, the variable \(x\) represents the number of women chosen for the two positions, with possible values 0, 1, or 2.
The probability for each value of \(x\) can be calculated using combinations:
  • \( p(x = 0) \): Both positions are filled with men. The formula yields \( \frac{3}{10} \) since there are 3 ways to pick 2 men from 3.
  • \( p(x = 1) \): One position is filled with a woman and the other with a man, yielding \( \frac{6}{10} \) as there are 6 ways to achieve this combination.
  • \( p(x = 2) \): Both positions are filled with women, just 1 way exists, giving \( \frac{1}{10} \).
These probabilities must all add up to 1, reinforcing the concept of a probability distribution, showing that all outcomes together encompass the entire sample space.
Histogram
A histogram is a graphical representation of data distribution, showing the frequency or probability of different outcomes. In our case, a probability histogram for \(x\) visually depicts how likely each number of women getting chosen is.
In our exercise, the probability histogram would have bars for each value of \(x\):
  • A bar at \(x = 0\) with a height of \(\frac{3}{10}\)
  • A bar at \(x = 1\) with a height of \(\frac{6}{10}\)
  • A bar at \(x = 2\) with a height of \(\frac{1}{10}\)
The height of each bar corresponds to the probability of each possible outcome. The histogram provides a clear, visual way to understand how the probability is distributed across the potential outcomes.
Random Variable
A random variable is a numerical outcome of a probabilistic event or experiment. It assigns numerical values to each of an experiment's outcomes. In the given exercise, the random variable \(x\) represents the number of women chosen for the two positions open.
This particular random variable can take on the values of 0, 1, and 2. These values represent the potential outcomes:
  • \(x = 0\): No women are chosen.
  • \(x = 1\): One woman and one man are chosen.
  • \(x = 2\): Both chosen are women.
Each of these outcomes is tied to its probability which we've calculated based on the number of ways each can occur out of the total possibilities. The random variable \(x\) is a fundamental element in constructing a probability model, aiding in analyzing processes or decisions involving uncertainty.

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Most popular questions from this chapter

The following information reflects the results of a survey reported by Mya Frazier in an \(A d\) Age Insights white paper. \({ }^{11}\) Working spouses were asked "Who is the household breadwinner?" Suppose that one person is selected at random from these 200 individuals. $$\begin{array}{lcccc} & \multicolumn{3}{c} {\text { Spouse or }} \\\& \text { You } & \text { Significant Other } & \text { About Equal } & \text { Totals } \\\\\hline \text { Men } & 64 & 16 & 20 & 100 \\\\\text { Women } & 32 & 45 & 23 & 100 \\\\\hline \text { Totals } & 96 & 61 & 43 & 200\end{array}$$ a. What is the probability that this person will identify his/herself as the household breadwinner? b. What is the probability that the person selected will be a man who indicates that he and his spouse/significant other are equal breadwinners? c. If the person selected indicates that the spouse or significant other is the breadwinner, what is the probability that the person is a man?

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