Overbooking math problem?

Question

The Greybeard Busing Company is assessing its overbooking policy for the Miami-Fort Meyers run. The number of customers who don't show up after reserving a ticket is uniformly distributed from 4 to 17. Tickets cost $88. If the bus is full, a passenger with a reservation is given passage on a rival bus line at a cost of $46. How may seats should Greybeard overbook?

Answer 1

I have taken a different approach. I look at the expected value of no-shows both greater and less than the selected overbook. These values come from statistical formulas and represent the actual values that would be achieved over a long period of time. The calcs are messy, so you may want to check for arithmetic or algebra errors:

Page 1 http://img248.imageshack.us/img248/231/overbook1fx1.png

Page 2 http://img258.imageshack.us/img258/9404/overbook2yo3.png

I get an optimum overbook of 13.

Answer 2

for every vacant seat the company loose 88$.
So if 17 seats are empty the company will not loose money for it. if 16 seats are empty the company will loose 88$ on the spare seat*P if this seat is free (P=1/14 in this case) if 15 seats are free the company will loose 88$*2*P... so y=(17-x)*P*88.
on the other hand: for every overbooked seat (over the 4th) the company loose 46$.
So if 4 seat are overbooked the company will not loose money.
if 5 seats ate overbooked the company will loose 46$*P if 56 seats are overbooked the company will loose 46$*P*2... so y = (x-4)*P*46
So you have two linear equations, I believe you can take it from here ;-)