Because all airline passengers do not show up for their...

Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is $0.10,$ and the passengers behave independently. (a) What is the probability that every passenger who shows up can take the flight? (b) What is the probability that the flight departs with empty seats?

Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is $0.10,$ and the passengers behave independently.
(a) What is the probability that every passenger who shows up can take the flight?
(b) What is the probability that the flight departs with empty seats?

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Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that represents the number of successes in a fixed number of independent trials, where each trial has two possible outcomes and the same probability of success. It is widely used to model events such as attendance or non-attendance in a series of trials.

Overbooking

Overbooking involves booking more resources than are available based on the expectation that a certain proportion of customers will cancel or not show up. This concept is applied in various industries like airlines, where statistical models help quantify the risk that the number of customers who attend will exceed capacity.

Cumulative Probability

Cumulative probability is the probability that a random variable is less than or equal to a certain value. It is calculated by summing the probabilities of all outcomes at or below that value, which is particularly useful for assessing the likelihood of events such as all arriving passengers fitting into a limited capacity.

Independence of Events

Independence of events means that the occurrence or non-occurrence of one event does not affect the occurrence of another. This assumption is essential in probabilistic models like the binomial distribution, where each trial is performed independently with the same probability of success.

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