Math Puzzle

/sci/ has been getting boring recently.  Let me attempt something new.  This is a math puzzle I heard about ages ago.

There are N boxes, each with a real number such that no two boxes have the same number.

The first box is opened so that you can see its contents.  You can a) choose this box or b) discard this box and open the next one [with which you repeat the procedure].  If you choose a), and the current box contains the largest number, then you win $1000.  If you choose b), then you will not be given the option of going back to the discarded box.  Of course, discarding all the boxes means that you lose by default.

What is the strategy to optimize your chances of winning, and what is this probability?

N=1, P=1
N=2, P=1/2
N=3, choosing the 1st box gives P=1/3, but if you discard the 1st box and choose the next largest box, you get P=3/6=1/2 > 1/3.
N>3, ...?

>>13
You're slow. If you discard the first box and choose the next box WHICH IS LARGER THAN THE FIRST, the probability is not 1/4.
Suppose A,B,C,D are boxes with A being the one containing the largest number, B being the next largest, and so on.

Here's the set of possible arrangements of the boxes, with "W" next to the winners by the strategy of discarding the first box and picking the next largest, and "L" next to the losers:

ABCD  L
ABDC  L
ACBD  L
ACDB  L
ADBC  L
ADCB  L
BACD  W
BADC  W
BCAD  W
BCDA  W
BDAC  W
BDCA  W
CABD  W
CADB  W
CBAD  L
CBDA  L
CDAB  W
CDBA  L
DABC  W
DACB  W
DBAC  L
DBCA  L
DCAB  L
DCBA  L

Sum up the W's and you get 11 wins, out of 24 possibilities. 11/24 > 1/4.

PS: To all the people saying "omg you don't know anything about the distribution": The entire fucking point of the exercise is that you gain knowledge about the distribution with each box you open, so how do you find the optimum strategy between opening every box (which gives you a 1/N chance of winning but perfect knowledge of the distribution) and choosing the first box (which leaves you with no knowledge of the distribution). No strategy is going to give you a "sure thing", but I've already shown in >>12 that some strategies are better than blind choosing.