Question

A tour bus can hold 20 tourists. Tickets costs $50 and are non-refundable. From previous experience, the tour operator knows there is a 5% chance of a no-show, independent of other ticket purchasers. The tour operator justifies that if there is a 5% (1 in 20) chance of a no show, he should expect that out of 20 tickets sold, 1 will not show up. In which case, the tour operator decides to sell 21 tickets. If a tourist shows up and the bus is full, the tour operator will refund their $50 for the ticket and give them another $150 as compensation. If the tour operator sells 21 tickets for a tour, what is their expected revenue? Is selling the extra ticket worth it?

Answer 1

There are 21 tickets sold.

Revenue from selling the 20 tickets when <=20 people show up = 20*50 = 1000

Revenue from selling 21 tickets when 21 people show up = 21*50 - (50+150) = 850

Probability of 21 people showing up(since the probability of each person showing up is independent of the other) = 0.95^21 = 0.341

Thus, probability of <=20 people showing up = 1-0.341 = 0.659

Expected revenue from 21 people showing up = 850*0.341= 290

Expected revenue from <=20 people showing up = 1000*.659= 659

Since expected revenue from selling 21 tickets is less than the expected revenue of selling 20 tickets, it is not worth selling the extra ticket.