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 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?

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

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 re: Table of Fair Amount | Comment 10 of 37 |
(In reply to Table of Fair Amount by Leming)

How does one get an expected value of 0.67 (or 2/3, presumably) if the strategy is to quit when one is even or ahead, until the cards run out.  For the 6 possible sequence of cards I get:

`--++ 0-+-+ 0-++- 0+--+ 1+-+- 1++-- !`

For a net win of 3 for every 6 games played making each one worh 1/2.

Or are you following a strategy different from quitting when one gets even or ahead?

What I get for the cases where A equals B or A is one less than B are:

` A  B   total won  plays  expected value      2  2         3        6 0.50000000000 2  3         2       10 0.20000000000 3  3        10       20 0.50000000000 3  4        10       35 0.28571428571 4  4        35       70 0.50000000000 4  5        42      126 0.33333333333 5  5       126      252 0.50000000000 5  6       168      462 0.36363636364 6  6       462      924 0.50000000000 6  7       660     1716 0.38461538462 7  7      1716     3432 0.50000000000 7  8      2574     6435 0.40000000000 8  8      6435    12870 0.50000000000 8  9     10010    24310 0.41176470588 9  9     24310    48620 0.50000000000 9 10     38896    92378 0.4210526315810 10     92378   184756 0.5000000000010 11    151164   352716 0.42857142857`

I note that the A=2, B=3 matches your previous result of 0.2, but not your new values.

DECLARE SUB permute (a\$)
DEFDBL A-Z
FOR a = 2 TO 10
FOR b = a TO a + 1
deckCt = 0
deck\$ = STRING\$(a, "+") + STRING\$(b, "-")
h\$ = deck\$
totWin = 0
DO
winnings = 0
FOR i = 1 TO LEN(deck\$)
IF MID\$(deck\$, i, 1) = "+" THEN
winnings = winnings + 1
ELSE
winnings = winnings - 1
END IF
IF winnings >= 0 THEN EXIT FOR
NEXT
totWin = totWin + winnings
deckCt = deckCt + 1
permute deck\$
LOOP UNTIL deck\$ = h\$
PRINT USING "## ## "; a; b;
PRINT USING " ########"; totWin; deckCt;
PRINT USING " #.###########"; totWin / deckCt

NEXT
NEXT

 Posted by Charlie on 2008-03-11 16:56:09

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