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Chapter 7: Problem 24

The City Housing Authority has received 50 applications from qualified applicants for eight low-income 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?

Short Answer

Expert verified
The probability that a specific qualified applicant will be selected for one of these apartments is approximately 0.0149, and the probability that two specific qualified applicants will be selected for apartments on the same side of town is 0.1.

Step by step solution

01

a. Probability of a specific qualified applicant being selected

Total applicants: 50 Total positions: 8 (3 on the north side and 5 on the south side) Step 1: Total number of ways to select 8 applicants from 50. This is a combination problem, as the order of selection does not matter. We use the formula: nCr = \( \dfrac{n!}{r!(n-r)!} \) where n = number of applicants and r = number of positions. Step 2: Calculate combinations. Total ways of selecting 8 applicants from 50: C(50, 8) = \( \dfrac{50!}{8!(50-8)!} \) Step 3: Calculate the probability. For a specific applicant to be selected, there are only 8 instances that their name is picked from the pool of all possible ways to fill the apartments. Probability = \( \dfrac{8}{C(50, 8)} \) Now, let's calculate the probability: a. Probability of a specific qualified applicant being selected C(50, 8) = \( \dfrac{50!}{8!(50-8)!} \) = 536878650 Probability of a specific applicant to be selected = \( \dfrac{8}{536878650} \) 鈮� 0.0149
02

b. Probability of two specific qualified applicants being selected for apartments in the same side of town

Step 1: Calculate combinations. There are two cases to consider: Case 1: Both selected applicants are assigned to the north side (3 positions available). C(3, 2) ways to assign the two specific applicants on the north side, and C(47, 6) ways to choose the remaining 6 applicants. Case 2: Both selected applicants are assigned to the south side (5 positions available). C(5, 2) ways to assign the two specific applicants on the south side, and C(47, 6) ways to choose the remaining 6 applicants. Step 2: Calculate the desired outcomes as follows: Desired outcomes = C(3, 2) * C(47, 6) + C(5, 2) * C(47, 6) Step 3: Calculate the probability. Probability = \( \dfrac{desired\_outcomes}{C(50, 8)} \) Now, let's calculate the probability: b. Probability of two specific qualified applicants being selected for apartments on the same side of town Desired outcomes = C(3, 2) * C(47, 6) + C(5, 2) * C(47, 6) = 3 * 10737573 + 10 * 10737573 = 53687865 Probability = \( \dfrac{53687865}{536878650} \) = 0.1 In conclusion, the probability that: a. A specific applicant will be selected for one of these apartments is approximately 0.0149. b. Two specific applicants will be selected for apartments on the same side of town is 0.1.

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Key Concepts

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

Probability theory
Probability theory is the mathematical framework we use to analyze uncertain events or outcomes. It helps us calculate the likelihood that a particular event will occur. In simple terms, probability measures how likely an event is to happen. It ranges from 0 to 1, where 0 means an event will not happen, and 1 means an event is certain to happen.

Understanding probability helps us make informed decisions in various fields like genetics, finance, and in every game of chance. The formula to calculate probability is given by:
  • Probability (P) = Number of favorable outcomes / Total number of possible outcomes
In the context of our exercise, we calculate the probability of selecting a specific applicant for an apartment by determining the ratio of favorable events (selecting that applicant) to the total possible events (different ways to fill the apartments). This requires understanding how combinations play a role in calculating total possible outcomes.
Lottery system
A lottery system is a method of randomly selecting winners from a pool of participants. In the context of housing allocation, like in our exercise, it helps decides who among a pool of applicants will be allotted the limited number of available apartments.

Lottery systems are widely used in various scenarios, such as:
  • Gambling games
  • School admissions
  • Allocation of scarce resources, like housing or tickets to events
This randomness ensures fairness, giving every participant an equal chance of winning, based on the rules set for the lottery. In our exercise, the lottery system randomly selects 8 applicants from a pool of 50, therefore requiring an understanding of probability and combinations to determine specific outcomes, like the selection of certain applicants.
Combination formula
The combination formula is a critical tool in combinatorics, which deals with counting and arranging objects. Unlike permutations, where order matters, combinations focus on the selection itself, not the sequence of selection.

The formula for combinations is:
\[C(n, r) = \frac{n!}{r!(n-r)!}\]
  • Where \( n \) is the total number of items,
  • \( r \) is the number of items to choose,
  • \( ! \) denotes factorial, the product of an integer and all the integers below it.
In our example, this formula calculates how many ways we can select 8 applicants from 50, without worrying about the order. We used combination theory to determine the total number of possible ways to select applicants for the apartments, and then compared these against specific scenarios, like a single applicant being chosen, or two applicants being chosen for the same side.

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Most popular questions from this chapter

In the game of blackjack, a 2 -card hand consisting of an ace and a face card or a 10 is called a blackjack. a. If a player is dealt 2 cards from a standard deck of 52 well-shuffled cards, what is the probability that the player will receive a blackjack? b. If a player is dealt 2 cards from 2 well-shuffled standard decks, what is the probability that the player will receive a blackjack?

In a study of the scientific research on soft drinks, juices, and milk, 50 studies were fully sponsored by the food industry, and 30 studies were conducted with no corporate ties. Of those that were fully sponsored by the food industry, \(14 \%\) of the participants found the products unfavorable, \(23 \%\) were neutral, and \(63 \%\) found the products favorable. Of those that had no industry funding, \(38 \%\) found the products unfavorable, \(15 \%\) were neutral, and \(47 \%\) found the products favorable. a. What is the probability that a participant selected at random found the products favorable? b. If a participant selected at random found the product favorable, what is the probability that he or she belongs to a group that participated in a corporate-sponsored study?

A study was conducted among a certain group of union members whose health insurance policies required second opinions prior to surgery. Of those members whose doctors advised them to have surgery, \(20 \%\) were informed by a second doctor that no surgery was needed. Of these, \(70 \%\) took the second doctor's opinion and did not go through with the surgery. Of the members who were advised to have surgery by both doctors, \(95 \%\) went through with the surgery. What is the probability that a union member who had surgery was advised to do so by a second doctor?

In a survey conducted in 2007 of 1402 workers 18 yr and older regarding their opinion on retirement benefits, the following data were obtained: 827 said that it was better to have excellent retirement benefits with a lower-than-expected salary, 477 said that it was better to have a higher-than- expected salary with poor retirement benefits, 42 said "neither," and 56 said "not sure." If a worker in the survey is selected at random, what is the probability that he or she answered that it was better to have a. Excellent retirement benefits with a lower-than-expected salary? b. A higher-than-expected salary with poor retirement benefits?

Suppose that \(A\) and \(B\) are mutually exclusive events and that \(P(A \cup B) \neq 0\). What is \(P(A \mid A \cup B)\) ?
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