Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The City Housing Authority has received 50 applications from qualified applicants for eight lowincome apartments. Three of the apartments are on the north side of town, and five are on the south side. If the apartments are to be assigned by means of a lottery, what is the probability that a. A specific qualified applicant will be selected for one of these apartments? b. Two specific qualified applicants will be selected for apartments on the same side of town?

Step-by-Step Solution

Answer

The probability that a specific qualified applicant will be selected for one of these apartments is \(\frac{2}{5}\). The probability that two specific qualified applicants will be selected for apartments on the same side of town is approximately 0.0000323.

Step 1: Problem Setup

To solve this problem, first, let's establish some notations and important data: - Total number of qualified applicants (A): 50 - Total number of apartments (P): 8 - North side apartments (N): 3 - South side apartments (S): 5 We'll determine probabilities for each part of the problem.

Step 2: Part (a): Probability for One Specific Applicant

To find the probability that a specific qualified applicant will be selected for one of these apartments, apply the combination formula to the total number of qualified applicants and apartments: The combination formula is \[C(n, k) = \frac{n!}{k!(n-k)!}\] In this case, - n = 50 (number of applicants) - k = 8 (number of apartments) First, find the total number of ways to choose 8 applicants from 50: Total combinations = \(C(50, 8) = \frac{50!}{8!(50-8)!} = 536878650\) Now, let's find the probability of one specific applicant being selected: The probability for one specific applicant is the number of ways they can be selected for any of the 8 apartments, divided by the total number of ways to choose 8 applicants from 50: Probability for one specific applicant: \[\frac{C(49, 7)}{C(50, 8)} = \frac{\frac{49!}{7!(49-7)!}}{\frac{50!}{8!(50-8)!}} = \frac{8}{50} = \frac{2}{5}\] Thus, the probability that a specific qualified applicant will be selected for one of these apartments is \(\frac{2}{5}\).

Step 3: Part (b): Probability for Two Specific Applicants on the Same Side

Now, let's find the probability that two specific qualified applicants will be selected for apartments on the same side of town: We will calculate the probability of both applicants being chosen on the north side or the south side and then add the probabilities. 1. Probability that both applicants are selected for apartments on the north side: - n = 48 (all other applicants) - k = 1 (remaining apartment on the north side) Combinations of north side apartments: \(C(48, 1) = 48\) 2. Probability that both applicants are selected for apartments on the south side: - n = 48 (all other applicants) - k = 3 (remaining apartments on the south side) Combinations of south side apartments: \(C(48, 3) = 17296\) Now, we add these probabilities: Total combinations for same side: 48 (north) + 17296 (south) = 17344 Finally, we find the probability by dividing the total combinations for the same side by the total number of ways to choose 8 applicants from 50: Probability for two specific applicants on the same side: \[\frac{17344}{536878650} \approx 0.0000323\] The probability that two specific qualified applicants will be selected for apartments on the same side of town is approximately 0.0000323.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and combining items in a set. It helps us understand how we can organize different elements according to defined rules. In the context of our original exercise, we used combinatorics to calculate the number of different ways to select applicants for housing.

The primary mathematical tool employed here is the combination formula, given by \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where:

• \(n\) is the total number of items—in this case, the qualified applicants.
• \(k\) is the number of items to choose—in this case, the apartments.
• \(!\)—called factorial—denotes the product of an integer and all the integers below it.

Using this formula helps determine the total number of ways to choose a subset of applicants from the total pool, facilitating the calculation of probabilities.

Applied Mathematics

Applied Mathematics involves using mathematical methods and reasoning to solve real-world problems. The original exercise is a perfect example of this. By breaking down the problem of housing allocation into smaller tasks, such as calculating specific probabilities, we bring mathematical attention to real-world scenarios.

In solving the exercise, we systematically identified the necessary data:

• Total number of qualified applicants (A) and apartments (P)
• The distribution of apartments between two town sides

We used the structure and discipline of mathematics to answer questions about fair housing allocation—a practical concern in urban planning.

Probability Theory

Probability Theory is the field of mathematics concerned with studying random events and outcomes. It helps us understand the likelihood of various outcomes in situations involving uncertainty. In this housing lottery exercise, we applied principles of probability theory to determine the chances of specific scenarios occurring.

Firstly, for a specific applicant to get an apartment, we computed the probability by considering the relationship between successful selections to total possibilities. We found this probability to be \(\frac{2}{5}\).

Secondly, to find the probability of two specific applicants getting apartments on the same side of town, we calculated distinct probabilities for each side and combined them. By analyzing different outcomes, probability theory helps us make informed predictions about real-world events.

Housing Allocation

Housing Allocation refers to the distribution of living spaces according to specific guidelines or criteria. The key challenge in housing allocation, as illustrated in the original problem, is ensuring a fair and transparent method of assigning limited housing resources to a larger group of applicants.

Considering factors such as the number of apartments and their locations, this exercise highlights two critical questions in allocation: the interest of individual fairness (e.g., probability that any single applicant gets an apartment) and collective fairness (e.g., probability of specific pairs being allocated nearby apartments).

By leveraging mathematics, particularly combinatorics and probability theory, we illustrated a method to approach such allocation issues. Mathematical techniques help ensure that such processes are not only fair but also based on transparent calculations, thus avoiding any element of bias or preference.