Question

In this exercise we examine the effects of overbooking in the airline industry. Ontario Gateway Airlines' first class cabins have 10 seats in each plane. Ontario's overbooking policy is to sell up to 11 first class tickets, since cancellations and no-shows are always possible (and indeed are quite likely). For a given flight on Ontario Gateway, there were 11 first class tickets sold. Suppose that each of the 11 persons who purchased tickets has a 20% chance of not showing up for the flight, and that the events that different persons show up for the flight are independent. The money for tickets to the passengers who didn’t show up are returned.

(a) What is the probability that at most 5 of the 11 persons who purchased first class tickets show up for the flight?

(b) What is the probability that exactly 10 of the persons who purchased first class tickets show up for the flight?

(c) Suppose that there are 10 seats in first class available and that the cost of each first class ticket is $1,200. (This $1,200 contributes entirely to profit since the variable cost associated with a passenger on a flight is close to zero.) Suppose further that any overbooked seat costs the airline $3,000, which is the cost of the free ticket issued the passenger plus some potential cost in damaged customer relations. (First class passengers do not expect to be bumped!) Thus, for example, if 10 of the first class passengers show up for the flight, the airline's profit is $12,000. If 11 first class passengers show up, the profit is $9,000. What is the expected profit from first class passengers for this flight?

(d) Suppose that only 10 first class tickets were sold. What would be the expected profit from first class passengers for this flight?

(e) People often travel in groups of two or more. Does this affect the independence assumption about passenger behavior?

Answer 1

The random variable representing the number of passengers who arrive out of 11 follows Binomial distribution with
The PMF of is

a) The probability that at most 5 of the 11 persons who purchased first class tickets show up for the flight is

R command below:
> pbinom(5,11,0.8)
[1] 0.01165421
b)  The probability that exactly 10 of the persons who purchased first class tickets show up for the flight is

R command below:

> dbinom(10,11,0.8)
[1] 0.2362232

c) The probability 10 or less of the first class passengers show up for the flight is

The probability that 11 first class passengers show up is

The expected profit from first-class passengers for this flight is

d) As given in part (c) of the question, the expected profit is