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Pick a card, any card.. (Posted on 2008-03-11) Difficulty: 3 of 5
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?

  Submitted by FrankM    
Rating: 2.7500 (4 votes)
Solution: (Hide)
The payoff conforms to the recursion relation:

W(A,B) = [1 + W(A-1,B)]A/[A+B] + [-1 + W(A,B-1)]B/[A+B]

Through substitution we can check that this is solved by

W(A,B) = Max[0, A(A+1-B)/(A+1)]

So a fair amount to pay would be A(A+1-B)/(A+1) (when A>=B). We should stop drawing as soon as there are more - cards than + cards remaining.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre(3): .. what went wrong .. to CharlieFrankM2008-03-25 15:15:57
Some Thoughtsre(2): .. what went wrong .. - to S HermanFrankM2008-03-25 14:51:20
re(2): .. what went wrong ..Charlie2008-03-25 11:36:53
re: .. what went wrong ..Steve Herman2008-03-25 11:08:25
Some Thoughts.. and on the eighth day, he began to scratch his head and wonder what went wrong ..FrankM2008-03-24 17:06:48
Hints/TipsEd B's questionsFrankM2008-03-24 16:33:41
re(2): A fair amount?Dej Mar2008-03-14 02:16:21
re(2): A fair amount?Charlie2008-03-13 21:33:33
re: Winning StrategyEigenray2008-03-13 20:57:28
re: A fair amount?ed bottemiller2008-03-13 19:49:55
A fair amount?Dej Mar2008-03-13 17:49:43
re: Winning StrategyCharlie2008-03-13 12:51:16
Winning Strategyed bottemiller2008-03-13 10:44:42
SolutionOptimal strategyEigenray2008-03-12 22:07:34
re: Incorrect assumptionCharlie2008-03-12 15:49:09
the posted official solution is wrongCharlie2008-03-12 15:44:49
re(2): Incorrect assumptionCharlie2008-03-12 15:20:55
re: Incorrect assumptioned bottemiller2008-03-12 11:59:35
re: Incorrect assumptionCharlie2008-03-12 11:18:43
re(3): When to play and when to cease drawing.Charlie2008-03-12 11:11:42
Some ThoughtsIncorrect assumptionFrankM2008-03-12 10:10:48
re(2): When to play and when to cease drawing.ed bottemiller2008-03-12 10:04:37
re: When to play and when to cease drawing.Charlie2008-03-11 23:15:11
re(5): What's the catch? - it's not that simpleCharlie2008-03-11 22:48:18
SolutionWhen to play and when to cease drawing.Leming2008-03-11 19:52:09
re(2): Table of Fair AmountLeming2008-03-11 19:33:11
re(4): What's the catch? - it's not that simpleed bottemiller2008-03-11 17:00:25
re: Table of Fair AmountCharlie2008-03-11 16:56:09
re(3): What's the catch? - it's not that simpleCharlie2008-03-11 16:09:16
SolutionTable of Fair AmountLeming2008-03-11 16:04:47
re(2): What's the catch? - it's not that simpleCharlie2008-03-11 15:52:31
re(3): solution--I thinkCharlie2008-03-11 15:46:22
re(2): solution--I thinked bottemiller2008-03-11 15:35:21
re: What's the catch? - it's not that simpleLeming2008-03-11 15:33:36
Some Thoughtsre: solution--I thinkLeming2008-03-11 15:16:43
What's the catch?ed bottemiller2008-03-11 15:11:44
Solutionsolution--I thinkCharlie2008-03-11 14:41:49
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