Binomial Distribution Problem – Airline Overbooking

Here’s the problem:

A Flight has 20 seats and enough demand to sell out all flights (not enough to justify buying bigger plane). All seats sell for $200. Probability a passenger with reservation shows up = p. The probability of “No show”=1-p. The occurrence of “Show/No show” is independent among passengers. Passengers who are turned away due to “overbooking” are given$240 (the purchase price plus a 20% penalty to airline)
The number of reservations made for each flight is chosen by airline (not random): n

The number of booked passengers who show up is a random variable: X

• What is the probability distribution of X?
• Write out the Revenues to the airline for a flight as a function of X
• Write out the expected Revenues to the airline
• What is the effect of increasing n on expected revenues for a given p?
• What is the effect of increasing p on expected revenues for a given n?
• When would it make no sense to overbook flights?

My thoughts:
I think the answer to the first part is that it is a binomial distribution (since we have two distinct outcomes and a constant and independent probability of success). However, I am facing problem in the remaining parts. Can someone please explain it to me? Thanks so much 🙂

What is the probability distribution of X?

Yes, $X \sim Binom(n, p)$. That is,

P(X=x) = \binom{n}{x}p^x(1-p)^x

Write out the revenues to the airline as a function of X.

If $X \leq 20$, then it is pure profit for the airlines. If $X > 20$, then the airline loses 40 dollars for each passenger over 20.

R(X) = 200X – \max[0, 240(X-20)]

Or…

R(X) =
\begin{cases}
200X & ;X \leq 20 \\
200(20) – 40(X-20) & ;X > 20
\end{cases}

Write out the expected value of R(X).

Remember that $E[R(X)] = \sum_{x=0}^n R(X)P(X=x)$. The math gets a bit tricky, but it can be done using R(X) as given above.

What is the effect of increasing n on expected revenues for a given p?

Eventually, it is going to decrease the expected revenue for the airline. Basically if the airline overbooks too often, they will lose a lot of money. However if $p$ is small, the expected value of the revenue may be optimal for some $n > 20$.

What is the effect of increasing p on expected revenues for a given n?

Again, it should decrease the expected revenue. If $n$ is fixed (and I’m assuming greater than 20), then increasing $p$ means the airline is more likely to have to refund somebodies money. You should be able to show this formally if you evaluate $E[R(X)]$.

When would it make no sense to overbook flights.

If $p=1$ (or in practice, if it is close to 1), then it makes no sense. Because they will always have to refund somebodies money.