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Chapter 5: Problem 25

Quota Problem: Motel Rooms The owners of a motel in Florida have noticed that in the long run, about \(40 \%\) of the people who stop and inquire about a room for the night actually rent a room. (a) Quota Problem How many inquiries must the owner answer to be \(99 \%\) sure of renting at least one room? (b) If 25 separate inquiries are made about rooms, what is the expected number of inquiries that will result in room rentals?

Short Answer

Expert verified
(a) At least 10 inquiries are needed. (b) The expected number is 10 room rentals.

Step by step solution

01

Understand the Problem Context

We are dealing with probabilities where the probability of renting a room from an inquiry is \( p = 0.4 \). Part (a) involves finding the number of inquiries needed to be 99% certain of success (renting at least one room). Part (b) requires calculating the expected number of successful room rentals from 25 inquiries.
02

Solve Part (a) 鈥� Define the Variables and Use the Complement Rule

For part (a), we define the probability of not renting a room from a single inquiry as \( q = 1 - p = 0.6 \). We want the probability of renting at least one room (success) to be 0.99. The probability of zero rentals in \( n \) inquiries is \( q^n \). Thus, the equation is \( 1 - q^n = 0.99 \). Simplifying, we get \( q^n = 0.01 \).
03

Calculate the Number of Inquiries in Part (a)

Converting the equation \( 0.6^n = 0.01 \) into a logarithmic form: \( n = \frac{\log(0.01)}{\log(0.6)} \). Using a calculator, we find: \( n = \frac{-2}{-0.2218} \approx 9.03 \). Rounding up since we can't have a fraction of an inquiry, \( n = 10 \).
04

Solve Part (b) 鈥� Calculate Expected Value

For part (b), the expected number of room rentals when 25 inquiries are made is calculated using the formula for expected value in a binomial distribution, \( E(X) = np \). Here, \( n = 25 \) and \( p = 0.4 \). Thus, \( E(X) = 25 \times 0.4 = 10 \).

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Key Concepts

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

Understanding Probability in Room Rentals
Probability is a way to measure how likely an event is to occur. In our motel room problem, the probability of renting a room when an inquiry is made is represented by the number 0.4 or 40%. This means that each time someone inquires about a room, there is a 40% chance they will decide to rent one.

In part (a) of the problem, we want to find out how many inquiries are needed to be 99% sure that at least one room is rented out. To solve this, we use the **Complement Rule**. The complement of an event is when that event does not happen. Here, the probability of not renting a room from an inquiry is 0.6, which is simply 1 - 0.4.

We aim to calculate how many inquiries ("n") make it practically impossible to have zero rentals. The formula used resembles this:
  • Choose the complement probability
  • Set your desired probability of the event occurring
  • Transform this into an equation
For our motel situation, we solve for a 0.99 success probability by forming and solving the geometric expression: \[ 1 - 0.6^n = 0.99 \] This formula helps determine the number of inquiries needed to reach a desired level of certainty.
Calculating Expected Value for Room Rentals
Expected value is a concept of finding the average outcome when an experiment is repeated many times. In terms of probability, it gives you the long-term average or mean of the outcomes when you try something multiple times.

In part (b) of our problem, we calculate the expected number of room rentals from 25 inquiries using expected value. The formula used is: \[ E(X) = np \] where "n" is the number of trials (inquiries in this case) and "p" is the probability of success (renting a room).

With 25 inquiries and a 40% rental probability, the expected value of rentals is:
  • Multiply the number of inquiries by the probability of success
  • Get an expected value of 10
This means that across many groups of 25 inquiries, on average, you expect 10 rooms to be rented.
The Role of Binomial Distribution in Room Rentals
A fundamental concept in probability and statistics is the **Binomial Distribution**, which models the number of successes in a series of trials where each trial is independent.

For our problem with the motel, each inquiry about a room can be thought of as a trial with a *success probability* of 0.4 (the room gets rented). The binomial distribution helps us understand and calculate specific outcomes, like how many rooms might get rented.

The probability mass function for a binomial distribution gives the likelihood of achieving "x" successes (i.e., room rentals) across "n" trials (i.e., inquiries). The binomial formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] This formula accounts for:
  • Total number of inquiries (n)
  • Number of rentals (successes, k)
  • Probability of renting from an inquiry (p)
  • Probability of not renting (1-p)
Understanding this distribution allows us to predict how many times a specific event may happen across different trials, such as estimating room occupancy based on past inquiries.

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Most popular questions from this chapter

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