An airline finds that 5 % of the persons making reservations...

An airline finds that $5 \%$ of the persons making reservations on a certain flight will not show up for the flight. If the airline sells 160 tickets for a flight that has only 155 seats, what is the probability that a seat will be available for every person holding a reservation and planning to fly?

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Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and the same probability of success. It is fundamental for problems where events have binary outcomes, such as a customer showing up or not showing up for a flight.

Cumulative Distribution Function (CDF)

The cumulative distribution function is used to determine the probability that a random variable will have a value less than or equal to a specified number. In the context of the binomial distribution, it allows us to calculate the likelihood that the number of successes (or customers showing up) does not exceed a certain threshold, which is crucial in assessing scenarios like overbooking.

Overbooking and Risk Management

Overbooking refers to the practice of selling more tickets or reservations than the available capacity, relying on the statistical probability of no-shows to manage risk. This strategy involves using probability theory to balance potential revenue against the risk of having to accommodate more customers than can be physically served, making it a key concept in operations and decision-making within industries like aviation.

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