This is a variation of the problem discussed some time back: Don't roll More. Just published the solution to the earlier problem. Thought it would be interesting to solve this problem taken once again from the book "Heard on The Street".
Problem: You have 52 playing cards (26 black and 26 red). You draw cards one by one. A red card pays you a dollar. A black card costs you a dollar. You can stop any point you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing expected payoff? Also, what is the expected payoff following the optimal rule?
Hint: Try the problem with 4 cards (2 red, 2 black) :)
Update(29/01/10): Question was incomplete. Added more information.
Solution: (Update (05/02/10)) Idea posted by Aman in comments!! Solution posted by me in comments!!
The problem/solution is very difficult and not so beautiful. Its not very mathematical though. Do this only if you have time and you are humble enough to accept defeat :P