An airline finds that 5 % of the persons who make...

An airline finds that $5 \%$ of the persons who make reservations on a certain flight do not show up for the flight. If the airline sells 160 tickets for a flight with 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 models the number of successes in a fixed number of independent trials with the same probability of success. It is used to analyze situations where each individual trial, such as whether a passenger shows up, has two possible outcomes (show or not show), and the overall outcome is the sum of these independent events.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a tool used to calculate the probability that a random variable takes a value less than or equal to a specific threshold. In scenarios involving the binomial distribution, the CDF helps in determining the likelihood that the number of successful outcomes, like the number of passengers that show up, does not exceed a certain limit.

Overbooking and Risk Management

Overbooking in industries like airlines involves selling more tickets than available seats based on the statistical expectation that a certain percentage of ticket holders will not show up. This concept requires an understanding of probability and risk management to balance revenue maximization with the likelihood of having more passengers than seats.

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