*To*: mathgroup@smc.vnet.net*Subject*: [mg12006] A comment on the Gambler's problem I recently posted*From*: "Daniel Sanders" <dman777@worldnet.att.net>*Date*: Fri, 17 Apr 1998 03:40:42 -0400*Organization*: AT&T WorldNet Services

Hi, Thank you all for posting or sending you solution to Galileo's gambling problem which I repeat before I add my comments. "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, a 9 seemed to come up less often a 10 supposedly in the experience of gamblers." A few of you observed and proved Mathematica-ly that the number of ten sums exceeds the number of nine sums in three throws of the dice. True enough, but the folks in Galileo's time didn't understand the concept of a probability space, and in particular an ordered n-tuple. The number of such permutations in the throw of three dice does sum to ten more frequently than it sums to nine, but the number of combinations is the same, and that is what the gambler's counted on. There are six combinations of 3-tuple that sum to nine and ten. Nine:{{1,2,6},{1,3,5},{1,4,4},{2,2,5},{2,3,4},{3,3,3}} Ten:{{1,3,6},{1,4,5},{2,2,6},{2,3,5},{2,4,4},{3,3,4}} And this another programming problem, that of a replacement rule that takes a finite probability space and kicks out a designated subset. In particular the 3-tuple set of the three dice and the number of combinations that sum to nine or ten. I was interested in the several ways that the last problem was answered, and I am working at understanding it all. Thanks.