*To*: mathgroup@smc.vnet.net*Subject*: [mg12012] Re: A Monte Carlo Simulation with loop structure*From*: Paul Abbott <paul@physics.uwa.edu.au>*Date*: Fri, 17 Apr 1998 03:40:48 -0400*Organization*: University of Western Australia

In response to Daniel Sanders, I wrote: >> a 9 seemed to come up less often a 10 supposedly in the experience of >> gamblers. >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}} Reading the question properly shows that I have not answered the question. One method (using an exhaustive search via pattern-matching) to compute the number of ways that sums of 9 and 10 can arise is as follows: In[1]:= ReplaceList[Table[Range[6],{3}], {{___, a_, ___},{___,b_, ___},{___,c_, ___}} :> {a, b, c} /; a + b + c == 9]//Length Out[1]= 25 In[2]:= ReplaceList[Table[Range[6],{3}], {{___, a_, ___},{___,b_, ___},{___,c_, ___}} :> {a, b, c} /; a + b + c == 10]//Length Out[2]= 27 Hence there are two more ways that sum to 10 (because summing to 9 includes the triple {3,3,3}). An exhaustive search can be avoided as follows: e.g., use Table to construct all ways with sum 9: Table[{i,j,9-i-j},{i,6},{j,Max[1,9-i-6],Min[6,9-i-1]}] 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 ____________________________________________________________________