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117 competition/games/craps.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

117 competition/games/craps.p


What are the odds in craps?

competition/games/craps.s

The game of craps:
There is a person who rolls the two dice, and then there is the house.
1) On the first roll, if a 7 or 11 comes up, the roller wins.
If a 2, 3, or 12 comes up the house wins.
Anything else is a POINT, and more rolling is necessary, as per rule 2.
2) If a POINT appears on the first roll, keep rolling the dice.
At each roll, if the POINT appears again, the roller wins.
At each roll, if a 7 comes up, the house wins.
Keep rolling until the POINT or a 7 comes up.

Then there are the players, and they are allowed to place their bets with
either the roller or with the house.

-----
My computations:

On the first roll, P.roller.trial(1) = 2/9, and P.house.trial(1) = 1/9.
Let P(x) stand for the probability of a 4,5,6,8,9,10 appearing.
Then on the second and onwards rolls, the probability is:

Roller:
                         ---                        (i - 2)
P.roller.trial(i) =      \   P(x)   *   ((5/6 - P(x))         *   P(x)
(i > 1)                  /
		         ---
		         x = 4,5,6,8,9,10
 
House:
                        ---                        (i - 2)
P.house.trial(i) =      \   P(x)   *   ((5/6 - P(x))         *   1/6
(i > 1)                 /
		        ---
		        x = 4,5,6,8,9,10

Reasoning (roller): For the roller to win on the ith trial, a POINT
should have appeared on the first trial (the first P(x) term), and the
same POINT should appear on the ith trial (the last P(x) term). All the in
between trials should come up with a number other than 7 or the POINT
(hence the (5/6 - P(x)) term).
Similar reasoning holds for the house.

The numbers are:
P.roller.trial(i) (i > 1) =
 
                (i-1)                 (i-1)                     (i-1)  
 1/72 * (27/36)      + 2/81 * (26/36)        + 25/648 * (25/36)
 
 
P.house.trial(i) (i > 1) =
 
                (i-1)                 (i-1)                     (i-1)  
 2/72 * (27/36)      + 3/81 * (26/36)        + 30/648 * (25/36)

-------------------------------------------------
The total probability comes to:

P.roller = 2/9   +   (1/18 + 4/45 + 25/198)  = 0.4929292929292929..
P.house  = 1/9   +   (1/9  + 2/15 + 15/99)  =  0.5070707070707070..

which is not even.
===========================================================================

Avinash Chopde (with standard disclaimer)
abc@unhcs.unh.edu, abc@unh.unh.edu {.....}!uunet!unh!abc

 

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