perplexus.info :: Probability : Pick a card, any card..

Pick a card, any card.. (Posted on 2008-03-11)

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
Puzzle Thoughts	K Sengupta	2024-05-08 22:09:02
re(3): .. what went wrong .. to Charlie	FrankM	2008-03-25 15:15:57
re(2): .. what went wrong .. - to S Herman	FrankM	2008-03-25 14:51:20
re(2): .. what went wrong ..	Charlie	2008-03-25 11:36:53
re: .. what went wrong ..	Steve Herman	2008-03-25 11:08:25
.. and on the eighth day, he began to scratch his head and wonder what went wrong ..	FrankM	2008-03-24 17:06:48
Ed B's questions	FrankM	2008-03-24 16:33:41
re(2): A fair amount?	Dej Mar	2008-03-14 02:16:21
re(2): A fair amount?	Charlie	2008-03-13 21:33:33
re: Winning Strategy	Eigenray	2008-03-13 20:57:28
re: A fair amount?	ed bottemiller	2008-03-13 19:49:55
A fair amount?	Dej Mar	2008-03-13 17:49:43
re: Winning Strategy	Charlie	2008-03-13 12:51:16
Winning Strategy	ed bottemiller	2008-03-13 10:44:42
Optimal strategy	Eigenray	2008-03-12 22:07:34
re: Incorrect assumption	Charlie	2008-03-12 15:49:09
the posted official solution is wrong	Charlie	2008-03-12 15:44:49
re(2): Incorrect assumption	Charlie	2008-03-12 15:20:55
re: Incorrect assumption	ed bottemiller	2008-03-12 11:59:35
re: Incorrect assumption	Charlie	2008-03-12 11:18:43
re(3): When to play and when to cease drawing.	Charlie	2008-03-12 11:11:42
Incorrect assumption	FrankM	2008-03-12 10:10:48
re(2): When to play and when to cease drawing.	ed bottemiller	2008-03-12 10:04:37
re: When to play and when to cease drawing.	Charlie	2008-03-11 23:15:11
re(5): What's the catch? - it's not that simple	Charlie	2008-03-11 22:48:18
When to play and when to cease drawing.	Leming	2008-03-11 19:52:09
re(2): Table of Fair Amount	Leming	2008-03-11 19:33:11
re(4): What's the catch? - it's not that simple	ed bottemiller	2008-03-11 17:00:25
re: Table of Fair Amount	Charlie	2008-03-11 16:56:09
re(3): What's the catch? - it's not that simple	Charlie	2008-03-11 16:09:16
Table of Fair Amount	Leming	2008-03-11 16:04:47
re(2): What's the catch? - it's not that simple	Charlie	2008-03-11 15:52:31
re(3): solution--I think	Charlie	2008-03-11 15:46:22
re(2): solution--I think	ed bottemiller	2008-03-11 15:35:21
re: What's the catch? - it's not that simple	Leming	2008-03-11 15:33:36
re: solution--I think	Leming	2008-03-11 15:16:43
What's the catch?	ed bottemiller	2008-03-11 15:11:44
solution--I think	Charlie	2008-03-11 14:41:49

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