Question

Overbooking flights

Eagle Air is a small commuter airline.  Each of their planes holds 15 people. Past records indicate that only 80% of people with reservations (tickets) do show up. Therefore, Eagle Air decides to overbook every flight. Suppose Eagle Air decides that it will accept up to 18 reservations per flight (18 is the maximum number of reservations per flight).

Demand for Eagle Air flights is so strong that 18 reservations are booked for every flight. Everyone knows how popular Eagle Air flights are, and so no one ever shows up without a reservation.

This is the first of 6 questions in this set.

a. Calculate the probability that on any given flight, at least one passenger holding a reservation will not have a seat.  Four decimals

b. What is the probability that there will be one or more empty seats? Four decimals

c. What is the probability that the first person who is bumped from a flight will not get on either of the next two flights? Assume that the number of “no-shows” is independent across flights. Also assume that the first person who is bumped has priority when an empty seat comes up on a subsequent flight.  Four decimals

d. What is the expected number of people who show up for a flight? Reminder: Everyone makes a reservation ahead; no one shows up without a reservation; a maximum of 18 reservations is accepted; and every flight has 18 reservations because Eagle Air is so popular.  One decimal

e. Suppose that each flight costs $1000 to run, considering all costs. If tickets are priced at $75 each, what is Eagle Air’s expected profit per flight? Assume that when sixteen or more people show up for a flight, any overbooked passengers wait until a seat becomes available, so Eagle Air ultimately gets the revenue from everyone who shows up.  One decimal

f. What is the standard deviation of the number of people who show up for a flight? Two decimals

Answer 1

The number of passengers (X) that show up for a flight, is a Binomial random variable with parameter n = 18, p = 0.8

a. At least one passenger will not have a seat, if X is at least 16. Hence, using a Binomial calculator the required probability is

b. One or emore seat will be empty, if X is at most 14

c. The person who is bumped will not be able to get on either of the next two flights, if at least 15 people show up on both these flights. Again, we calculate

Since n-shows are independent across flights, required probability using product rule of independent events is

d. The number of people who show up is a Binomial random variable, so it is a simple application of the expected value formula for a Binomial RV

e. Since the expected arrivals calculated in previous part is less than the available seats, so in the long run everyone wil lboard the flight. Hence, the expected profit is simply total revenue minus total costs.

f. Again using the standard result on Binomial RV