Can you beat the Casino?
There is a stack of \(100\) blank cards. The casino dealer then proceeds to secretly write any positive integer on each of the cards. Once he is done, the deck is handed over to you, numbers facing down. You may shuffle cards as you desire. You can then count and turn over as many cards as you wish.
If the last card that you choose to flip turns out to be the card with the highest number in the deck, you win. Otherwise you lose.
Each game costs \(100\) bucks to play. If you win the game, you get \(500\) bucks; otherwise you lose only the \(100\) bucks you paid to play. The dealer writes down a new set of numbers every game.
Do you think you should play this game?
Using OPTIMAL strategy, what is the expected value of your profit (or loss designated by negative sign) after playing \(400\) games ?