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 ?.
Best Answer
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:
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 $q$.