Question

Four union men, two from a minority group, are assigned to four distinctly different one-man jobs, which can be ranked in order of desirability. a. Define the experiment. b. List the simple events in \(S\). c. If the assignment to the jobs is unbiased - that is, if any one ordering of assignments is as probable as any other-what is the probability that the two men from the minority group are assigned to the least desirable jobs?

Short answer

Answer: The probability is 0.25 or 25%.

Step-by-step solution

Define the experiment

The experiment consists of assigning four union men, including two from a minority group, to four distinctly different one-man jobs that can be ranked in order of desirability. The union men are randomly assigned to the jobs.

List the simple events in S

Let's denote the union men as A, B (minority group), C, D, and the jobs as 1 (most desirable), 2, 3, and 4 (least desirable). There are 4! (4 factorial) ways to assign the union men to the jobs. This means there are 24 possible assignments in the sample space \(S\). The sample space \(S\) contains the following arrangements (A, B, C, D represent Union men, and the numbers 1, 2, 3, and 4 represent the jobs): 1. A-1, B-2, C-3, D-4 2. A-1, B-2, C-4, D-3 3. A-1, B-3, C-2, D-4 4. A-1, B-3, C-4, D-2 5. A-1, B-4, C-2, D-3 6. A-1, B-4, C-3, D-2 7. A-2, B-1, C-3, D-4 8. A-2, B-1, C-4, D-3 9. A-2, B-3, C-1, D-4 10. A-2, B-3, C-4, D-1 11. A-2, B-4, C-1, D-3 12. A-2, B-4, C-3, D-1 13. A-3, B-1, C-2, D-4 14. A-3, B-1, C-4, D-2 15. A-3, B-2, C-1, D-4 16. A-3, B-2, C-4, D-1 17. A-3, B-4, C-1, D-2 18. A-3, B-4, C-2, D-1 19. A-4, B-1, C-2, D-3 20. A-4, B-1, C-3, D-2 21. A-4, B-2, C-1, D-3 22. A-4, B-2, C-3, D-1 23. A-4, B-3, C-1, D-2 24. A-4, B-3, C-2, D-1

Calculate the probability that the two men from the minority group are assigned to the least desirable jobs

We need to find the probability that men from the minority group (B and C) are assigned to jobs 3 and 4, which are the least desirable jobs. We can count the number of arrangements in which this occurs and divide it by the total number of arrangements (24). There are 6 arrangements in which B and C are both assigned to jobs 3 and 4: 1. A-1, B-3, C-2, D-4 2. A-1, B-3, C-4, D-2 3. A-1, B-4, C-2, D-3 4. A-1, B-4, C-3, D-2 5. A-2, B-3, C-1, D-4 6. A-2, B-3, C-4, D-1 The probability that the two men from the minority group are assigned to the least desirable jobs is therefore the ratio of these arrangements to the total number of arrangements: Probability = (Number of favorable arrangements) / (Total number of arrangements) = 6 / 24 = 0.25 Thus, the probability that the two men from the minority group are assigned to the least desirable jobs is 0.25 or 25%.

Key concepts

Sample Space

In probability theory, the concept of a sample space is fundamental. A sample space, usually denoted by the symbol \(S\), encompasses all possible outcomes of a particular experiment or random trial. In the given scenario, our experiment involves assigning four union men, including two from a minority group, to four different jobs that range in desirability. The sample space here includes every possible arrangement of the men to the jobs.

With four men to be assigned to four jobs, we can calculate the total number of possible assignments using factorial notation. The number of arrangements is given by \(4!\), which translates to \(4 \times 3 \times 2 \times 1 = 24\). This tells us there are 24 unique ways the men can be assigned to the jobs. Each set of assignments represents a simple event within our sample space.

Simple Events

Simple events are individual outcomes within the sample space that cannot be broken down further. Each simple event in our scenario represents a unique way the union men can be assigned to the jobs. Since there are 24 different ways to arrange the four men, there are 24 simple events. An example of a simple event would be assigning "A" to the most desirable job, "B" to the second most desirable, and so on.

The list of these 24 simple events can help us understand the nature of possibilities within our sample space. Recognizing each of these outcomes as individual simple events is key for calculating probabilities of more complex events involving these basic outcomes.

Unbiased Assignment

Unbiased assignment implies that every possible arrangement of the union men to jobs is equally likely. In other words, there is no preference or bias that would make some assignments more probable than others. This assumption of equal likelihood is crucial when dealing with probability as it allows us to calculate probabilities by simple ratio.

In context, with 24 possible assignments, each has a probability of being chosen of \(\frac{1}{24}\). This means if any man is just as likely to land any job as another, we can say the assignment process is fair and unbiased. For probability calculations, this foundational idea simplifies the process considerably, allowing us to assume each arrangement in the sample space has the same chance of occurring.

Minority Group

The term "minority group" in this context refers to two of the four men who belong to a distinct category, different from the other union men. The role of the minority group in this exercise is crucial as the problem involves calculating probabilities related to them specifically being assigned to certain jobs.

Understanding how many members of the minority group exist and identifying them helps in calculating specific probabilities, such as them being assigned to less desirable jobs. For these kinds of problems, the distinction often forms the basis of the probability questions we are trying to solve, such as ensuring fair job allocation or analyzing the impact of being in a minority group within the selection process.

Desirability Ranking

The jobs in the exercise are ranked by desirability, which means they are not valued equally by the union men. Assigning desirability rankings adds a layer of complexity because jobs are not interchangeable from the workers' perspectives. The rankings inform which jobs are considered more or less favorable.

In the exercise, jobs are ranked from 1 to 4, with 1 being the most desirable and 4 the least. Understanding the ranking helps in analyzing probabilities of specific events, such as determining which men might be assigned to more desirable jobs in various arrangements or, as in this scenario, to less desirable ones. This introduces a common real-world factor into probability problems — the fact that not all outcomes are equally preferable, which can influence how we think about assignments and fairness.