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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?

No Solution Yet Submitted by FrankM    
Rating: 2.7500 (4 votes)

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re: Incorrect assumption | Comment 19 of 37 |
(In reply to Incorrect assumption by FrankM)

Only ed bottemiller continues to believe one should refuse to play or stop playing when there is an equality in the number of + and - cards remaining.  I initally thought that, but leming corrected me. My extended table below,

 b: 0      1    2     3     4     5     6     7     8     9     10    11    12 
 1.00  0.50  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  2.00  1.33  0.67  0.20  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  3.00  2.25  1.50  0.85  0.34  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  4.00  3.20  2.40  1.66  1.00  0.44  0.07  0.00  0.00  0.00  0.00  0.00  0.00
  5.00  4.17  3.33  2.54  1.79  1.12  0.55  0.15  0.00  0.00  0.00  0.00  0.00
  6.00  5.14  4.29  3.45  2.66  1.91  1.23  0.66  0.23  0.00  0.00  0.00  0.00
  7.00  6.13  5.25  4.39  3.56  2.76  2.01  1.34  0.75  0.30  0.00  0.00  0.00
  8.00  7.11  6.22  5.35  4.49  3.66  2.86  2.11  1.43  0.84  0.36  0.05  0.00
  9.00  8.10  7.20  6.31  5.43  4.58  3.75  2.95  2.21  1.52  0.92  0.43  0.10
10.00  9.09  8.18  7.28  6.39  5.52  4.66  3.83  3.04  2.30  1.61  1.00  0.50
11.00 10.08  9.17  8.26  7.35  6.46  5.59  4.74  3.91  3.12  2.38  1.69  1.08
12.00 11.08 10.15  9.24  8.32  7.42  6.54  5.66  4.81  3.99  3.20  2.46  1.77

shows that if there is 1 +, play should stop if there are 2 or more -'s. If there are 2 +'s, stop if there are 4 or more -'s (where the zeros begin in the expected values). If there are 3 +'s, stop if there are 5 or more -'s. If there are 4 +'s stop if there are 7 or more -'s.

The table is of course recursive, rather than closed-form, but does give the expected value for a given A and B, such as 8.32 when A=12 and B=4.


  Posted by Charlie on 2008-03-12 11:18:43
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