Problem 2 - Applied Probability

Question

Two positions for campus jobs are open. Two sophomores, three juniors, and three seniors apply. It is decided to select two at random (each possible pair equally likely). Let \(X\) be the number of sophomores and \(Y\) be the number of juniors who are selected. Determine the joint distribution for the pair \(\\{X, Y\\}\) and from this determine the marginals for each.

Step-by-Step Solution

Answer

The joint distribution for \(\{X, Y\}\) will indicate the probabilities of selecting 0 or more sophomores/juniors. The marginals for each \(X\) and \(Y\) are calculated by summing probabilities across the joint distribution for fixed values of \(X\) and \(Y\), respectively.

Step 1 - Calculate total number of applicants

The total number of applicants is the sum of sophomores, juniors, and seniors applying for the positions. Calculate this sum to use in further calculations.

Step 2 - Calculate the total number of ways to select two applicants

Use the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\) to determine the total number of ways to select two applicants out of the total calculated in Step 1.

Step 3 - List all possible pairs for \(\{X, Y\}\)

List all the combinations of sophomores and juniors that can be selected. These pairs are \(\{0,0\}\), \(\{1,0\}\), \(\{0,1\}\), \(\{2,0\}\), \(\{0,2\}\), \(\{1,1\}\).

Step 4 - Calculate the number of ways each pair \(\{X, Y\}\) can occur

Use the combination formula to calculate the number of ways to select each possible pair of sophomores and juniors, considering the total number of each class.

Step 5 - Determine the probability for each pair \(\{X, Y\}\)

Divide the number of ways to select each pair by the total number of ways to select two applicants to find the probability for each pair combination.

Step 6 - Determine the marginal distributions for \(X\) and \(Y\)

Sum the probabilities across the joint distribution for a fixed value of \(X\) to find the marginal distribution for \(X\) and sum the probabilities for a fixed value of \(Y\) for the marginal distribution for \(Y\).

Key Concepts

Marginal Distribution

When studying the concept of joint distribution in probability theory, the marginal distribution becomes a vital aspect to grasp. It involves calculating the probabilities of various outcomes for a single random variable irrespective of the outcomes of other random variables. To put it simply, marginal distribution provides us with the probability distribution of one variable disregarding the presence of another.

To determine the marginals in our exercise with the campus job applicants, one would sum-up the joint probabilities over the values of one variable to get the marginal probability for the other. For instance, to get the marginal distribution for the number of sophomores (X), we add up the probabilities of all possible combinations where X occurs, irrespective of the values of Y. This step is crucial, as it distills the complex joint distribution information into a more manageable form, focusing only on one variable at a time.

Combination Formula

Understanding the Basics

Central to solving problems involving random selection from a group is the combination formula. Utilized in our campus job selection problem, this formula enables us to calculate the number of different ways we can pick a set number of items from a larger set, where the order of selection is irrelevant.

Mathematically, it's written as
\(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n\) represents the total number of items to choose from, \(k\) is the number of items to be chosen, and \(!\) denotes a factorial, the product of all positive integers up to that number.

For our example, where we are selecting two applicants from eight, the combination formula helps calculate the total number of possible pairs. It is the groundwork for understanding the various selection combinations before diving into their respective probabilities.

Probability Theory

At the core of such exercises is probability theory, a fundamental branch of mathematics that deals with the likelihood of different outcomes. It provides the mathematical framework to quantify and analyze randomness and uncertainty. Probability theory allows us to calculate numerical measures that convey the chance of occurrence of various possible events.

In our textbook solution, probability theory comes into play when we calculate the likelihood of selecting particular groups of students for the campus jobs. After using the combination formula to find the total number of ways to form pairs of applicants, we divide this by the individual probabilities of selecting each possible student pairing. This notion of determining 'how likely' an event is, underpins all of probability theory and is the cornerstone of the calculations conducted in the given exercise.

Random Selection

Role in Probability

Random selection is a principle that guarantees each member of a set has an equal opportunity of being chosen. It's a fair way to make a choice without any bias or pattern. In our example, random selection implies that each pair of students has the same chance to be selected for the campus jobs.

Implementing a random selection method in probability problems, like the one with the campus job applicants, is crucial because it aligns with the concept of equal likelihood, which is a basic assumption in calculating probabilities for simple events. The random selection criterion makes sure that the probabilities we calculate reflect the real-world situation where indeed every potential applicant pair has the same shot at being chosen.