Question

Overbooking a Boeing 767-300 A Boeing 767-300 aircraft has 213 seats. When someone buys a ticket for a flight, there is a 0.0995 probability that the person will not show up for the flight (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). How many reservations could be accepted for a Boeing 767-300 for there to be at least a 0.95 probability that all reservation holders who show will be accommodated?

Short answer

A total of 241 reservations could be accepted for the flight, keeping in mind the probability of at least 0.95 that all reservation holders who show will be accommodated.

Step-by-step solution

Given information

There is a 0.0995 probability that a person will not show up for the flight. There are 213 seats in the specified aircraft.

Conversion of p-value to z-score

It is given that the probability that a person will not show up for the flight is equal to 0.0995.

It is required to determine the total number of reservations that could be accepted to accommodate at least 95% of the people who show up.

In terms of z-score, the value of the z-score corresponding to the probability of 0.95 is equal to 1.645.

Conversion of z-score to sample value

Let X denote the number of people who do not show up after making a reservation.

The probability that a person will not show up after making a reservation is equal to p=0.0995.

Here, n=213.

The following formula is used to convert the z-score to the required sample value:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ x=\mu +z\sigma \end{gathered}$$

The mean value is computed below:

$$\begin{gathered} \mu =np \\ =213\times 0.0995 \\ =21.1935 \end{gathered}$$

The standard deviation is computed below:

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{np(1-p)} \\ =\sqrt{213\times 0.0995\times (1-0.0995)} \\ \approx 4.3686 \end{gathered}$$

Thus, the value of x is equal to:

$$\begin{gathered} x=\mu +z\sigma \\ =21.1935+1.645\times 4.3686 \\ =28.37 \\ \approx 28 \end{gathered}$$

Now, the total number of reservations that can be accepted, including the number of reservations that are not fulfilled, is computed below:

$213+28=241$

Therefore, 241 reservations could be accepted for the flight, keeping in mind at least a 0.95 probability that all reservation holders who show will be accommodated.