# Overbooked flight probability problem

1. Feb 24, 2009

### bluebear19

Airlines often overbook flights. You are a manager selling 40 tickets on a flight with a capacity of 35. You estimate the probability of a given passenger NOT showing up is 0.2.

I already calculated some parts of the problem:
1) how many passengers do you expect to show up = (.8)(40) = 32
2) what is the probability that more than 35 will show up = .0759

Problem:
Suppose a ticket sells for $300. What is the maximum cost per passenger who must be "bumped" (compensation paid to ticketed passenger who can't take the overbooked plane) such it is worth (expected value) to sell 40 tickets versus 35. (For simplicity assume that you still get your$300 from anyone who purchased a ticket and doesn't show up)

Thank you in advance!!

2. Feb 24, 2009

### CRGreathouse

Let P(n) be the probability that exactly n people show up. The expected number of people you must compensate is
N = P(36) + 2P(37) + 3P(38) + 4P(39) + 5P(40)

For overbooking to 40 to be better than selling 35, NC < \$300 * 5, where C is the total amount you must compensate passengers who are bumped.