Next week, Super Discount Airlines has a flight from New York to Los Angeles that will be booked to capacity. The airline knows from past history that an average of 25 customers (with a standard deviation of 15) cancel their reservation or do not show for the flight. Revenue from a ticket on the flight is $125. If the flight is overbooked, the airline has a policy of getting the customer on the next available flight and giving the person a free round-trip ticket on a future flight. The cost of this free round-trip ticket averages $250. Super Discount considers the cost of flying the plane from New York to Los Angeles a sunk cost.
Let p = probability of no shows being x or fewer (so if we overbook x + 1 seats instead of x, we lose $250 with probability p, and gain $125 with probability 1-p)
Up to what value of p is it profitable to overbook one more seat?