You sit down with a well mixed deck containing A cards marked "+" and B cards marked "—". You may draw cards from this deck as long as you want, i.e., you can stop playing at any point. Each time you draw a + card you are given $1 and each time you draw a — card you have to pay $1. Cards are
not replaced after having been drawn.
What would be a fair amount to pay for the right to play (i.e., what is the expected payoff) and under what circumstance should a player cease drawing?
If you do not know the counts of A and B, you should NEVER play this game.
If you do know the counts in advance, you should play only if A is greater than B, and the cost to play is less than A$ - B$. (It is not clear whether "the fair amount to pay for the right to play" means a fee for EACH card drawn, or for the right to play until you stop or the pack is exhausted (since cards are not replaced). It's a dumb wager in any case.
If you decide to play (i.e. A and B known and A > B), then you will know the total T number of cards involved. You should stop whenever the remaining A cards are fewer than or equal to the remaining B cards (and you have been keeping track of the total of each type).
In either case, get a life!
What's the catch?
