Probability - Airplane overselling tickets

Question

Few days ago, I came across a question for probability in one of the interview.

Question :

The same small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.1, independent of other passengers. The average fare paid by a passenger who succeeds in boarding is $100. A passenger who shows up but cannot board is given 200 dollars and a free flight later on.

Given that the airline can sell 30 + m tickets, how should the airline set m to maximize expected revenue?

I did not get how to solve this probability question. Can somebody please suggest a approach to solving this kind of questions ?

Answer 1

Assuming that the free flight is equivalent to selling a ticket for a future flight which is then not sold to somebody else and there is no refund, then it is worth $\$100$ and the compensation has a total cost to the airline of $\$300$.

The total gross revenue for a flight is $\$100\times(30+m)$ and the expected compensation is $\displaystyle \sum_{i = 1}^m \$300\times i \times {30+m \choose 30+i} 0.9^{30+i} 0.1^{m-i}$ , with the expect net revenue being the difference between these.

There are various ways of finding the maximum expected net revenue. One would be to work out the values for sensible values of $m$: since $30/0.9 \approx 33.3$, it might worth looking at $m=3$ as this will have the plane often full or almost full. Then look at $m$ marginally more or less, i.e. $m=2$ or $4$.