I'm currently stuck on the following problem where we know that for an airline it is a 7% probability that a passenger will not meet up for departure. So to get a better use of the plane capacity the airline overbook the tickets. The plane for this task has 243 seats.
The thing is that the airline loses 1000\$ for each non-used seat in the plane. If a passenger does not get a spot because of the overbooking, the passenger will be compensated with 4000\$. We want to find an expression for the expected loss if we take $n$ orders. And by using the expression/graph we want to find the number of orders such that we minimize the expected loss. The numerical answer is given, which is 258 orders which results a loss of $5564\$$.
I understand that this will be a binomial distribution if we define a stochastic variable $X$ which says how many passengers that really meet ups for departure... but how do I use this fact to get the desired expression? If we do not overbook, i.e. only take 243 orders, the expected loss is to be $243 \cdot 0.07 \cdot 1000 = 17010 \$$, but if I try to do the same for $n = 258$ orders I do not get the same answer as the solution(which would be $5564\$$), why could I do that for $n = 243$, but not for $n= 258$?