Lotus Blossom

LOTUS BLOSSOM HOTEL

The Lotus Blossom Hotel has a total of 300 rooms. Like many hotels, and most major airlines, the Lotus Blossom Hotel has adopted a policy of overbooking in an effort to maximize usage of its available lodging capacity.

Past data indicate that 12% of the people holding reservations cancel the reservation, or simply do not show up at the hotel on a given night, while 88% of the people with reservations show up to keep their reservation. Lotus Blossom management believes that it is a good approximation to the real world to assume that each reservation is kept or cancelled, independent of the situation with any other reservation.

Lotus Blossom management is concerned with profitability, and filling otherwise vacant rooms, but it also concerned with having to tell people with a reservation that there is no room available for them. The latter situation greatly reduces the goodwill that Lotus Blossom enjoys, even though Lotus Blossom gets the person a room at another close-by and good quality hotel. Thus, Lotus Blossom management is trying to decide how to “trade off” having fewer vacancies (which means more overbooking) and not losing goodwill (which means less overbooking). It knows that if there were no overbooking, no goodwill would ever be lost, but there would be a relatively large number of vacancies each evening, hurting profitability. Yet, if too much overbooking takes place, there will be a lot of people turned away from the hotel, and a relatively large amount of goodwill will be lost.

To help Lotus Blossom management, answer the following:

1) What is the largest number of reservations Lotus Blossom can make, and still have at least a 95% chance that everyone who shows up that evening will have a room available?

2) If Lotus Blossom books the number of reservations determined in question 1, what is the probability that at least 95% of the rooms are occupied on a given evening?

3) Repeat questions 1 and 2 if we...