Expected Revenue earned by a tour operator

Question

A tour operator has a bus that can accomodate 20 tourists. The operator knows that a tourist may not show up, so he sells 21 tickests. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up . If a tourist shows up and a seat is not available, the tour operator has to pay 100 to that tourist. What is the expected revenue of the tour operator ?.

My working: Since the tickets are non refundable the tour operator has an initial revenue of 21 x 50 = 1050. The revenue which is the random variable here, will vary according to the probability of the 21st person showing up. Thus the expected revenue would be equal to 1050 x 0.02 + 950 x 0.98 = 952. However my answer is incorrect. Could somebody help me out ?.

Answer 1

Your mistake is in thinking that the operator must return money if the 21st person showed up, but in fact, that's not true. For example, if the first person doesn't show up, then the operator will not be returning any money no matter if the 21st person shows up or not.

In fact, you have two possibilities:

• All $21$ people show up, in which case the opeator earns $950$. The probability of this is $p$.
• At least one person doesn't show up, in which case the operator earns $1050$. The probability of this is $q$.

Certainly, one of these two possibilities happens, and they can't happen at the same time (they are disjoint), so you already know that $p+q=1$ and you only need either $p$ or $q$.

It should also be much easier to calculate $p$ directly compared to $q$.

Answer 2

I'm assuming that the operator is selling 21 tickets, and not 20 like you say in your problem statement.

There are two different outcomes here. If all 21 tourists show up, one of them won't have a seat so the operator will have to pay 100, otherwise the operator doesn't pay anything.

The probability for all 21 tourists to show up is the probability that all tourists don't not show up, i.e. $P=(1-0.02)^{21}\approx0.6543$. The initial revenue is $21*50=1050$ no matter what, so the expected revenue is: $1050-100 * P \approx984.57$.

On average the tour operator loses money by selling the 21st ticket.