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# Re: A Monte Carlo Simulation with loop structure
*To*: mathgroup@smc.vnet.net
*Subject*: [mg11999] Re: A Monte Carlo Simulation with loop structure
*From*: Paul Abbott <paul@physics.uwa.edu.au>
*Date*: Fri, 17 Apr 1998 03:40:35 -0400
*Organization*: University of Western Australia
*References*: <6gr6oo$8n4@smc.vnet.net>
Daniel Sanders wrote:
> I owe the problem to J. Laurie Snell from his book, "Introduction to
> Probability".
> In the early 1600s, Galileo was asked to explain the fact that,
> although the number of triples of integers from 1 to 6 with sum 9 is
> the same as the number of such triples with sum 10, when three dice are
> rolled,
One way to check this is to use ReplaceList:
In[1]:= ReplaceList[Range[6], {___, a_, ___, b_, ___, c_, ___} :>
{a, b, c} /; a + b + c == 9]
Out[1]= {{2, 3, 4}, {1, 3, 5}, {1, 2, 6}}
In[2]:= ReplaceList[Range[6], {___, a_, ___, b_, ___, c_, ___} :>
{a, b, c} /; a + b + c == 10]
Out[2]= {{2, 3, 5}, {1, 4, 5}, {1, 3, 6}}
> a 9 seemed to come up less often a 10 supposedly in the experience of
> gamblers.
> Can you suggest native Mathematica code not emulating a procedural
> language for a Monte Carlo simulation experiment that runs many times
> keeping track of the proportions of 9s and 10s? Thanks in advance,
> I'm interested in the looping mechanism.
We roll three dice, with each die as
In[3]:= die := Random[Integer, {1, 6}]
10000 times as follows:
In[4]:= Apply[Plus, Table[die, {10000}, {3}], 1];
and then count the 9s and 10s:
In[5]:= {Count[%, 9], Count[%, 10]}
Out[5]= {1181, 1288}
We can use Table to do looping:
In[6]:= Table[With[{rolls=Apply[Plus, Table[die, {10000}, {3}], 1]},
{Count[rolls, 9], Count[rolls, 10]}],{10}]
Out[6]= {{1120, 1242}, {1176, 1194}, {1134, 1259}, {1155, 1306},
{1135, 1297}, {1110, 1256}, {1224, 1215}, {1171, 1248},
{1129, 1245}, {1155, 1277}}
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul@physics.uwa.edu.au AUSTRALIA
http://www.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
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