Airline Reservations In Chapter 6, we learned that the proportion of passengers who miss a flight for which they have a reservation is 0.0995.


a. Suppose a flight has 290 reservations. What is the probability that 25 or more passengers will miss the flight?

Verified step by step guidance

1. Identify the type of distribution to model the number of passengers who miss the flight. Since each passenger either misses the flight or not, and the probability of missing is constant, this situation can be modeled using a binomial distribution with parameters \(n = 290\) (number of trials) and \(p = 0.0995\) (probability of success, i.e., missing the flight).
2. Define the random variable \(X\) as the number of passengers who miss the flight. Then, \(X\) follows a binomial distribution: \(X \sim \text{Binomial}(n=290, p=0.0995)\).
3. The problem asks for the probability that 25 or more passengers miss the flight, which can be written as \(P(X \geq 25)\). This is equivalent to \$1 - P(X \leq 24)$.
4. Calculate \(P(X \leq 24)\), the cumulative probability that at most 24 passengers miss the flight. This can be done using binomial cumulative distribution function (CDF) tables, a calculator with binomial functions, or statistical software.
5. Finally, subtract the cumulative probability from 1 to find the desired probability: \(P(X \geq 25) = 1 - P(X \leq 24)\). This gives the probability that 25 or more passengers miss the flight.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, each passenger either misses the flight (success) or not (failure), with a known probability. It is used to calculate probabilities of a certain number of passengers missing the flight out of total reservations.

Normal Approximation to the Binomial

When the number of trials is large, the binomial distribution can be approximated by a normal distribution to simplify calculations. This requires checking conditions like np and n(1-p) being sufficiently large. The approximation allows using z-scores and standard normal tables to find probabilities for events like 25 or more passengers missing the flight.

Calculating Tail Probabilities

Tail probabilities refer to the likelihood of observing values at the extreme ends of a distribution, such as 25 or more passengers missing the flight. Calculating these involves summing probabilities from a cutoff point to the maximum or using complement rules with cumulative distribution functions. This helps assess rare or significant events in the context of the problem.