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?
(In reply to re: When to play and when to cease drawing.
Charlie and Leming appear obdurate in what I take to be a confused reinterpretation of the puzzle as posed. They seem to be intent on providing tables of odds which would only be relevant if one were forced to play and continue playing (which clearly is not the case). They seem to focus on only the first phrase of the task: "what would be a fair amount to pay for the right to play," more or less ignoring the second: "under what circumstances should a player cease drawing". They correctly state that the fair amount is A - B, and surely see that this is negative if B is greater than A -- but provide no rationale for starting or continuing when B is greater.
I would suppose the answer is simply: quit (or don't start at all) when your expectations are negative. (Perhaps C and L realize this, and are just trying to find something interesting to do with what seems a trivial puzzle.)
I am willing to admit I have sometimes overlooked a critical factor in a puzzle, but I still don't see the reasoning they are pursuing. Either (1) the potential player knows in advance how many cards are A, and how many B, or (2) he lacks this information. Perhaps they suppose that with each draw you are NOT informed of the outcome, the payout or loss being announced only when the mark decides to quit or exhausts the pack -- but clearly the puzzle says "each time you draw" you are given a dollar or pay in a dollar.
If (2), he has NO way to assess the "fair amount" either to begin play, or at any point to continue. Hypothetical tables of probabilities are of no use, unless one has SOME ground to believe in the distribution of A and B (perhaps the game would be unfair if there were NO cards of either one or the other). Perhaps the potential mark could first observe someone else play, and count the As and Bs -- or something of the sort -- but that goes beyond the problem as stated.
The logic of playing at any point when there are RANDOMLY more B than A remaining totally escapes me (and we are told by the problem setter -- not by the con who might lie -- that this is supposed to be an honest game with a "well-mixed deck".
Since I know there are people who do play lotteries or drawings, even though there is a negative expectation, one would need to introduce a putative psychological value for the fun of playing, the excitement value or whatever (perhaps the net goes to a worthwhile charity) -- in which case the actual "fair amount to pay," is purely subjective, and the answer to the second part of the question would be he should quit when he gets tired of being a sucker.