Problem 25 - An Introduction to Mathematical Statistics and Its Applications

Question

Each year a college awards five merit-based scholarships to members of the entering freshman class who have exceptional high school records. The initial pool of applicants for the upcoming academic year has been reduced to a "short list" of eight men and ten women, all of whom seem equally deserving. If the awards are made at random from among the eighteen finalists, what are the chances that both men and women will be represented?

Step-by-Step Solution

Answer

The probability that both men and women will be represented is the number of all possible combinations excluding those where only men or women are chosen, divided by the total number of combinations.

Step 1: Total Combinations

Find the total number of ways to select 5 candidates out of 18. This calculation can be done using the combination formula which is \( C(n, k) = \frac{n!}{k!(n-k)!} \). In our case, \( n = 18 \) and \( k = 5 \). Using these values, we can derive the total number of combinations.

Step 2: Calculate Non-Inclusive Cases

Identify the situations in which only men or only women are awarded. The situations include either choosing 5 from 8 men or 5 from 10 women, and we should calculate each separately and add them. The same combination formula from Step 1 can be used here to determine the number of such combinations.

Step 3: Calculate Desired Probability

Finally, subtract the number of cases where only men or only women are awarded from the total number of combinations. Then divide this result by the total number of combinations to get the probability. The formula for this is \( P(A) = \frac{n(A)}{n(S)} \) where \( n(A) \) is the number of favorable outcomes and \( n(S) \) is the total number of outcomes.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, among others. In the context of probability, combinatorics is essential for determining the number of possible outcomes in various scenarios.

For example, if we want to know how many different ways we can arrange a group of people into seats at a table, or in the case of our exercise, how many different groups of scholarship awardees can be selected from a larger pool of applicants, combinatorics provides the tools to calculate these numbers without having to list all possible arrangements or selections one by one.

Combination Formula

The combination formula is a fundamental part of combinatorics used to determine the number of ways to choose a subset of items from a larger set when the order of selection does not matter. The formula is denoted 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 choose, and \( ! \) denotes factorial, the product of all positive integers up to that number.

In our scholarship example, you use the combination formula to calculate the total possible ways to select 5 candidates from 18 finalists (\( n = 18 \) and \( k = 5 \)). This formula is invaluable in many areas of statistics and probability, providing a straightforward method to calculate what would otherwise be an intractable number of possible combinations.

Favorable Outcomes in Probability

In probability, favorable outcomes are the specific results of an experiment or random trial that we're interested in. To calculate the probability of an event, we need to know both the number of favorable outcomes and the total number of possible outcomes in the sample space.

Take our scholarship example. If we want to find the probability that our selection will include both men and women, we consider 'favorable outcomes' to be the cases where there is at least one man and one woman in the group. To calculate this, we must first identify all combinations that do not meet our criteria (only men or only women) and then subtract these from the total number of combinations to obtain the number it is of 'favorable outcomes'.

Total Outcomes in Probability

Total outcomes in probability, also known as the sample space, are all the possible results of an experiment or random process. For accurate probability calculations, it's crucial to count all of these outcomes without omission or duplication. The total number of outcomes serves as the denominator when computing the probability of an event.

In the scholarship scenario, the total outcomes represent all the different ways we can choose 5 candidates from the pool of 18 finalists. This forms the basis for the probability calculation, as any specific set of candidates chosen (the favorable outcomes) is a subset of this larger set. Once we have the total and the number of favorable outcomes, we can use the probability formula to find the chances of selecting a mixed group of men and women for the scholarships.