Sayure Praers, a small-plane, short-haul airline, has found...

Sayure Praers, a small-plane, short-haul airline, has found that approximately $5 \%$ of all persons holding reservations on a certain flight do not show up. If the plane holds 50 passengers and Sayure takes reservations for 53 (this is called overbooking), what is the probability that every passenger who arrives on time for the flight will have a seat? (Assume there are no walk-ins.)

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes (success or failure) with a constant probability of success. It is crucial for calculating the probability of a given number of arrivals (or no-shows) when each reservation independently results in a show or no-show.

Bernoulli Trials

Bernoulli trials are experiments that result in only one of two possible outcomes, typically labeled as success and failure, with fixed probabilities for each outcome. This concept is fundamental since each passenger’s decision to show up can be considered an independent Bernoulli trial.

Cumulative Probability

The cumulative probability measures the likelihood that a random variable takes on a value less than or equal to a certain number. In the context of the binomial distribution, it is used to calculate the overall probability that the total number of successes (e.g., passengers showing up) does not exceed a specified threshold.

Overbooking in Probability

Overbooking involves accepting more reservations than available capacity based on the probability of some reservations being unused. This concept uses probability models to balance the risk of having more arrivals than seats available against the likelihood that not all reservations will be fulfilled.

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