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Chapter 5: Problem 6

From long experience a landlord knows that the probability an apartment in a complex will not be rented is \(0.10 .\) There are 20 apartments in the complex, and the rental status of each apartment is independent of the status of the others. When a minimum of 16 apartment units are rented, the landlord can meet all monthly expenses. Which probability is more relevant to the landlord in terms of being able to meet expenses: the probability that there are exactly four unrented units or the probability that there are four or fewer unrented units? Explain.

Short Answer

Expert verified
\(P(X \leq 4)\) is more relevant, as it includes cases where 16 or more apartments are rented.

Step by step solution

01

Define the Problem

We need to determine which probability is more relevant to the landlord: the probability that exactly four units are unrented, or the probability that four or fewer units are unrented, knowing that unrented probability for each apartment is 0.10.
02

Model the Situation Using a Binomial Distribution

This is a binomial problem with number of trials \(n = 20\) (since there are 20 apartments), and probability of success (unrented) \(p = 0.10\). A success here means an apartment is not rented.
03

Calculate Probability for Exactly Four Unrented Units

Using the binomial probability formula, \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\), compute for \(k = 4\):\[ P(X = 4) = \binom{20}{4} (0.10)^4 (0.90)^{16} \]
04

Calculate Probability for Four or Fewer Unrented Units

Use cumulative probability for 0 to 4 unrented units:\[ P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \]For each value of \(X\), use \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\).
05

Determine Which Probability is More Relevant

Since meeting monthly expenses requires at least 16 rented apartments, the landlord would want fewer than 5 unrented apartments (\(X \leq 4\)) to maximize the number of rented apartments. Thus, \(P(X \leq 4)\) is more relevant.

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics and mathematics that expresses the likelihood of an event happening. In simple terms, it's a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Understanding probability helps us make informed decisions and predictions.

When determining probabilities, it is important to define what constitutes a 'trial' and a 'success.' In our case, a trial refers to the rental status of one apartment, and a 'success' occurs if the apartment is not rented. We know the probability of an apartment not being rented is 0.10.

In the rental scenario, the probability helps the landlord estimate how likely it is that a certain number of apartments will go unrented. By breaking this down, the landlord can plan their finances more effectively, ensuring they meet their expense needs by renting a sufficient number of units.
Rental Probability
Rental Probability in this context refers to the specific probability calculation related to the landlord's properties.

To solve the problem, we employ the binomial distribution model since we are dealing with independent trials (each apartment can be rented or not), a fixed number of trials (20 apartments), and two possible outcomes (rented or not rented).

In the exercise, we need to consider two probabilities:
  • The first is the probability that exactly four apartments are not rented, which we calculate using the binomial probability formula as: \[P(X = 4) = \binom{20}{4} (0.10)^4 (0.90)^{16}.\]
  • The second is the probability that four or fewer apartments are not rented. This is found by summing up individual probabilities for having 0, 1, 2, 3, and 4 unrented units, denoted as \[P(X \leq 4).\]
These calculations give the landlord critical insights about the rental outcomes.
Cumulative Probability
Cumulative Probability is a concept that provides the probability that a random variable is less than or equal to a certain value. It is particularly useful when assessing a range of outcomes, rather than focusing on a single possibility.

In our scenario, calculating cumulative probability helps the landlord determine the odds of having up to four unrented apartments. This is crucial for ensuring the landlord can meet their monthly expenses. They need at most four apartments to stay unrented to comfortably cover all expenses, implying at least 16 should be rented out.

To find this, you sum up all the probabilities starting from 0 to 4: \[P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).\]

This cumulative approach is more relevant for real-world decision-making, as in this case, it guides the landlord towards ensuring a profitable rental situation.

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Most popular questions from this chapter

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