I was thinking about a gambling problem when I wanted to genralised it.I dont know to what extend the question is known.A guy take a regaular deck of card and shuffle it. Then he takes half of it. The shuffling has been perfect (ie the position of the card in the original deck is totaly random).Even if you dont know how much card there is in this deck, you know the likeliness of the composition.If you see a lot of black card passing, you know that a red card is more likely to come even you never know for sure if it is a black card or a red card next whatever happened before. (that is why I cutted the deck in two.....)Know, the guy make you bet on a 1:1 game black or red card. How to play to maximise your gain ?An easy answer would be to wait till the end to have the more information you can, however i am sure you can have a better expectation than that. All the more, how much are you going to risk ? If you bet all your equity at once and lose, you cant play it again.I am not looking for the best expectation but the best return (ie you need to survive).

Your game rules are not clear, but it sounds like a simplification of blackjack with card counting. Kelly criterion looks like what you're looking for.

Well, I didnt said it right away but I am trying here to do an extension of Kelly criterion. Can you be more specific with what is not clear ?I mean, you need a little bit more here that just the conditnal probablity to make Kelly owrk , or am I wrong ?