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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.6.22
Chapter 6, Problem 6.6.22

Overbooking a Boeing 767-300 A Boeing 767-300 aircraft has 213 seats. When someone buys a ticket for a flight, there is a 0.0995 probability that the person will not show up for the flight (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). How many reservations could be accepted for a Boeing 767-300 for there to be at least a 0.95 probability that all reservation holders who show will be accommodated?

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Step 1: Define the problem using a binomial distribution. The number of reservations accepted can be modeled as a binomial random variable, where each reservation has a probability of 0.0995 of not showing up and a probability of 0.9005 of showing up. The goal is to find the maximum number of reservations (n) such that the probability of accommodating all passengers who show up is at least 0.95.
Step 2: Establish the relationship between the number of reservations (n), the number of seats (213), and the probability of accommodating all passengers. The probability of accommodating all passengers is the probability that the number of passengers who show up is less than or equal to 213. This can be expressed as P(X ≤ 213), where X is the binomial random variable representing the number of passengers who show up.
Step 3: Use the binomial probability formula to calculate P(X ≤ 213). The binomial probability formula is given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient, p is the probability of showing up (0.9005), and k is the number of passengers who show up. To find P(X ≤ 213), sum the probabilities for all values of k from 0 to 213.
Step 4: Use a statistical software or calculator to compute the cumulative probability P(X ≤ 213) for different values of n. Start with n slightly greater than 213 and increase it incrementally until P(X ≤ 213) is at least 0.95. This iterative process helps identify the maximum number of reservations that can be accepted while ensuring the desired probability.
Step 5: Verify the result by checking that P(X ≤ 213) is indeed at least 0.95 for the chosen value of n. If necessary, refine the value of n to ensure the condition is met. Once verified, this value of n represents the maximum number of reservations that can be accepted.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability of No-Show

The probability of no-show refers to the likelihood that a ticket holder will not attend the flight. In this scenario, it is given as 0.0995, meaning there is a 9.95% chance that a passenger will not show up. Understanding this probability is crucial for calculating how many reservations can be made while ensuring that the aircraft can accommodate all passengers who do show up.
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Binomial Distribution

The binomial distribution is a statistical model that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it can be used to model the number of passengers who show up for the flight, allowing us to calculate the probability of accommodating all reservation holders based on the number of tickets sold.
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Cumulative Probability

Cumulative probability is the probability that a random variable takes on a value less than or equal to a certain value. In this case, we need to calculate the cumulative probability of accommodating all passengers who show up, ensuring it is at least 0.95. This involves summing the probabilities of all scenarios where the number of passengers showing up does not exceed the number of available seats.
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