Question

United flight 15 from New York’s JFK to San Francisco uses a Boeing 757-200 with 180seats. Because some people with tickets don’t show up, United willoverbook by selling more than 180tickets. If the flight is not overbooked, the airline will lose revenue due to empty seats, but if too many tickets are sold and some passengers are denied seats, the airline loses money from the compensation that must be given to bumped passengers. Assume that there is a 0.905 probability that a passenger with a ticketwill show up for the flight. Also assume that the airline sells200tickets for the 180seats that are available.

Answer 1

There is a 29.27% probability that the flight is overbooked. This is not an unusually low probability. So it does seem too high so that changes must be made to make it lower.

Explanation:

For each passenger, there are only two outcomes possible. Either they show up for the flight, or they do not show up. This means that we can solve this problem using binomial distribution probability concepts.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And is the probability of X happening.

A probability is said to be unusually low if it is lower than 5%.

For this problem, we have that:

There are 200 reservations, so .

A passenger consists in a passenger not showing up. There is a .0995 probability that a passenger with a reservation will not show up for the flight. So .

Find the probability that when 200 reservations are accepted for United Flight 15, there are more passengers showing up than there are seats available.

X is the number of passengers that do not show up. It needs to be at least 18 for the flight not being overbooked. So we want to find , with . We can use a binomial probability calculator, and we find that:

There is a 29.27% probability that the flight is overbooked. This is not an unusually low probability. So it does seem too high so that changes must be made to make it lower.