Monte Carlo Simulation of the Craps Dice Game

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Monte Carlo Simulationof the Craps Dice Game

• Sencer Koç

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Basics

• The player rolling the dice is the "shooter".
  Shooters first throw in a round of Craps is
  called the Come Out roll. If you roll a 7 or 11,
  you win and the round is over before it started.
• If you roll a 2, 3, or 12 that's a Craps and you
  lose again, it's over before it started.
• Any other number becomes the Point. The purpose
  of the Come Out roll is to set the Point, which
  can be any of 4, 5, 6, 8, 9 or 10.

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Objective

• The basic objective in Craps is for the shooter
  to win by tossing the Point again before he
  tosses a 7. That 7 is called Out 7 to
  differentiate it from the 7 on the Come Out roll.
• If the Point is tossed, the shooter and his
  fellow bettors win and the round is over. If the
  shooter tosses Out 7, they lose and the round is
  over.
• If the toss is neither the Point nor Out 7, the
  round continues and the dice keep rolling.

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Craps Game
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Questions

• What is the probability that the roller wins?
• Note that this is not a simple problem.
• The probability of win at later rolls depends on
  the point value, e.g., if the point is 8,
  P(8)5/36 (2,6,3,5,4,4,5,3,6,2) and if
  the point is 10, P(10)1/36 (5,5).
• How many rolls (on the average) will the game
  last?

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Simulation and MATLAB

• We will simulate the craps game on a computer
  using MATLAB.
• The command rand(1)
• Uniformly distributed between 0 and 1
• round(rand(1,2)60.5)simulates a single roll of
  a pair of dice.
• The following MATLAB code simulates M games.

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MATLAB Code
M 10 Number of simulations lost 0 win
0 gamelength zeros(1,M) disp(' win
loss gamelength') data zeros(10000,3) for
k1M stop 0 curr_gamelength 1
x sum(round(rand(1,2)60.5)) if (x7)
(x11) win win1 stop 1 elseif
(x2) (x3) (x12) lost lost1
stop 1 else point x end
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MATLAB Code (Continued)
while stop curr_gamelength
curr_gamelength1 x sum(round(rand(1,2)
60.5)) if (x7)
lostlost1 stop 1 elseif (xpoint)
win win1 stop 1 end
end gamelength(k) curr_gamelength
disp(sprintf('8.0f 8.0f 8.0f '...
,win, lost, curr_gamelength)) data(k,)
win lost curr_gamelength end
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Sample Simulation
Program Output
Trial win loss gamelength 1
0 1 2 2 1
0 7 3 1 0
3 4 0 1 2
5 1 0 5 6
0 1 10 7 1
0 8 8 0 1
2 9 0 1 4
10 1 0 1
Summary Data
Total over 10 simulations win loss
gamelength 5 5 4.4
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Generation of Useful Data

• Remember we are using a random number generator!
• We also need a larger sample size. We will run
  the problem for many more games!
• Let us now try 1000 games and divide wins and
  loss by M to determine the probabilities
  (comment out the disp commands to suppress output
  of data).
• Also evaluate the mean of gamelengths.
• Our point estimators are
• P(win) data(1000,1)/1000
• P(loss) data(1000,2)/1000
• L mean(data(,3))

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Sample Simulations (M1000)
Trial P(win) P(loss) L 1
0.479 0.521 3.646 2
0.496 0.504 3.774 3
0.497 0.503 3.351 4
0.467 0.533 3.450 5
0.512 0.488 3.386 6
0.485 0.515 3.435 7
0.480 0.520 3.476 8
0.494 0.506 3.591 9
0.492 0.508 3.466 10
0.495 0.505 3.312

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Now make N1000 such trials!

• This will take some time (about 5 minutes) since
  the MATLAB code is written for readability and
  not for speed!
• You may use the following command line
• reszeros(1000,3)for ii11000 mcscraps
  res(ii,)win/1000 lost/1000 mean(gamelength)en
  d

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Histograms
Histogram of P(win) for 1000 trials.
Histogram of L for 1000 trials.
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Confidence Intervals
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Exact
The exact value of probability of win can be
calculated by using the theory of Markov Chains
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Final Note

• The game length is an exponentially distributed
  random variable, however its mean over N1000
  trials (as shown before) is approximately
  normally distributed (Central Limit Theorem)