*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 ____________________________________________________________________