Hotel generally has reservations for all 300 rooms and people trying to make a r

Question

Hotel generally has reservations for all 300 rooms and people trying to make a room reservation are told the hotel is full. Hotel allows customers to cancel their room reservation without charge up until check-in time. This policy results in the hotel having less than 100% occupancy even when they have 300 reservations. By analyzing historical data Hotel has determined that 7% of reservations will be canceled and cancelations are independent of each other. It costs Hotel $24,000 per day to operate and each room cost $100 per night.

a) What is the expected occupancy rate on a day when Hotel has 300 reservations?

b) What is the expected daily profit on a day when Hotel has 300 reservations?

c) What is the probability Hotel will have 95% or more occupancy on a day when they have 300 reservations?

Explanation / Answer

(a) Expected occupancy rate on a day when hotel has 300 reservations = 300 * 0.93 = 279

(b) expected daily profit on a day when Hotel has 300 reservations = 100 * 279 - 24000 = $ 3900

(c) Here as there are seven percent of reservations will be cancled then as we can say it is binomial distribution where n = 300 and p = 0.93

Now 95% or more occupancy means = 300 * 0.95 = 285 or more reservations

but we can also approximate the binomial distribution into normal as n is high.

so here, if X is the number of occupancy out of 300 is

E(X) = 300 * 0.93 = 279

Var(X) = sqrt (300 * 0.93 * 0.07) = 4.4193

Pr(X >= 285) = BIN (X > =285 ; 279 ; 4.42) = NORM (X > 284.5 ; 279 ; 4.4193)

Z = (284.5 - 279)/4.4193 = 1.2445

So,

Pr(X >= 285) = 1 -Pr(Z < 1.2445) = 1 - 0.8933 = 0.1067

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