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(2): What's the catch? - it's not that simple by Charlie)
I built a spreadsheet for calculating the Fair Amount. From that point on, the itterative equation is:
f[A,B]= (Probability of drawing A) * (profit of drawing A plus profitability of f[A-1,B]) + (Probability of drawing B) * (Loss of drawing B plus profitability of f[A,B-1])
f[A,B] = A/(A+B)*(1+f[A-1,B]) + B/(A+B)*(-1+f[A,B-1])
where f[0,B] = 0 and f[A,0] = A
A / B 0 1 2 3 4 5
0 0.00 0.00 0.00 0.00 0.00 0.00
1 1.00 0.50 0.00 0.00 0.00 0.00
2 2.00 1.33 0.67 0.20 0.00 0.00
3 3.00 2.25 1.50 0.85 0.34 0.00
4 4.00 3.20 2.40 1.66 1.00 0.44
5 5.00 4.17 3.33 2.54 1.79 1.12
Edited for clarity and format and content
Edited on March 11, 2008, 4:29 pm
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Posted by Leming
on 2008-03-11 16:04:47 |
Table of Fair Amount
