A small commuter airline flies planes that can seat up to 8...

A small commuter airline flies planes that can seat up to 8 passengers. The airline has determined that the probability that a ticketed passenger will not show up for a flight is 0.2. For each flight the airline sells tickets to the first 10 people placing orders. The probability distribution for the number of tickets sold per flight is shown in the accompanying table. For what proportion of the airline's flights does the number of ticketed passengers showing up exceed the number of available seats? (Assume independence between the number of tickets sold and the probability that a ticketed passenger will show up.) $$ \begin{array}{lccccc} \hline \text { Number of tickets } & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Probability } & 0.25 & 0.35 & 0.25 & 0.10 & 0.05 \\ \hline \end{array} $$

A small commuter airline flies planes that can seat up to 8 passengers. The airline has determined that the probability that a ticketed passenger will not show up for a flight is 0.2. For each flight the airline sells tickets to the first 10 people placing orders. The probability distribution for the number of tickets sold per flight is shown in the accompanying table. For what proportion of the airline's flights does the number of ticketed passengers showing up exceed the number of available seats? (Assume independence between the number of tickets sold and the probability that a ticketed passenger will show up.)
$$
\begin{array}{lccccc}
\hline \text { Number of tickets } & 6 & 7 & 8 & 9 & 10 \\
\hline \text { Probability } & 0.25 & 0.35 & 0.25 & 0.10 & 0.05 \\
\hline
\end{array}
$$

Key Concepts

Probability Distributions

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It can be used to model various random processes, such as the number of tickets sold for a flight or the number of passengers who show up. Understanding probability distributions is central to predicting outcomes and making decisions under uncertainty.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two outcomes (success or failure). In contexts like passenger no-shows, where each ticketed passenger independently shows up with a certain probability, the binomial distribution provides a way to calculate the likelihood of a given number of successes (i.e., passengers showing up).

Law of Total Probability

The law of total probability is used to compute the probability of an event by considering all the different mutually exclusive scenarios that could lead to that event. When faced with hierarchical or layered uncertainties (such as varying numbers of tickets sold and independent probabilities of showing up), this principle allows one to sum over the conditional probabilities of the event across different cases.

Independence of Events

Independence of events means that the outcome of one event does not affect the outcome of another. In this context, the assumption that the number of tickets sold is independent of whether a ticketed passenger shows up enables separate treatment of these probabilities, simplifying the analysis and application of probability rules such as the binomial distribution and the law of total probability.

Overbooking and Capacity Constraints

Overbooking refers to the practice of selling more tickets than there are available seats, anticipating that some ticket holders will not show up. In operations management and revenue optimization, understanding the probability that demand (passengers showing up) will exceed capacity is essential to manage risks associated with overbooking and to determine appropriate policies for ticket sales and cancellations.

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