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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Incorrect assumption | Comment 21 of 37 |
(In reply to re: Incorrect assumption by ed bottemiller)

Leming and I have been going on the assumption that the "fair amount" is paid once at the beginning of the game, rather than at each draw of a card.

Based on the table (extended farther to the right than I've posted), the following are the largest numbers of -'s that would be allowable to continue playing with a given number of +'s:

# +'s/max -'s to keep trying
  1   1
  2   3
  3   4
  4   6
  5   7
  6   8
  7   9
  8  11
  9  12
 10  13
 11  14
 12  15

As a sampling, the following program simulates starting with 8 +'s and 10 -'s.

DEFDBL A-Z
DIM hitOn(12)
DATA 1,3,4,6,7,8,9,11,12,13,14,15
FOR i = 1 TO 12: READ hitOn(i)
PRINT i; hitOn(i)
: NEXT
RANDOMIZE TIMER
totWinnings = 0
DO
 winnings = 0

 plus = 8: minus = 10
 DO
  r = INT(RND(1) * (plus + minus) + 1)
  IF r <= plus THEN
    winnings = winnings + 1
    plus = plus - 1
  ELSE
    winnings = winnings - 1
    minus = minus - 1
  END IF
 LOOP UNTIL minus > hitOn(plus) OR plus = 0

 totWinnings = totWinnings + winnings
 gameCt = gameCt + 1
 PRINT totWinnings, gameCt, totWinnings / gameCt
LOOP

I halted it at 109,459 trials of this game using the above strategy for stopping and the accumulated winnings were 39,124, for an average winnings per full game of 0.35743. The table previously presented shows, to the limit of accuracy of that table, 0.36.


  Posted by Charlie on 2008-03-12 15:20:55
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