THE DOCTRINE OF CHANCES. 17
we shall find the number of times 8 may come up is = to 6 + 5+4 + 3 + 2 + 1=21 ; thus there are 15 times for 7, 10 times for 6, 6 times for 5, 3 times for 4, 1 for 3, 25 times for 9, 27 times for 10, 27 times for 11, 25 for 12, 9 for 13, 15 times for 14, 10 times for 15, 6 times for 16, 3 times for 17, 1 only for 18. Thus 10 and 11 are with three dice the most advantageous to bet in favour of, the odds in favour of their being thrown being 27 to 216, or 8 to 1.
By this method we may determine the numbers most likely to be thrown with any number of dice.
It will be obvious from the above, how essential it is to know the number of combinations of which any number of dice are susceptible, in order to avoid accepting disadvantageous bets, which is but two often the fate of those who do not reflect that all chances are in some degree submitted to mathematical analysis.
Two dice, as we have just observed, being taken together, form twenty-one numbers, and considered separately, will give thirty-six different combinations. Of the 21 coups which may be thrown with two dice, the first 6 are doublets, and can only be thrown once, as the 2 sixes, &c. &c. The 15 other coups, on the contrary, have each two combinations, the aggregate number of the whole being 36. The odds, therefore, of the caster throwing a given doublet are 1 to 35 ; and again, of his throwing an indeterminate one, 1 to 5 ; and 1 to 17 that he throws 6 and 4, seeing that this point gives him two chances against 34.
But it is not the same with the number of points of two dice joined together ; the combination of their chances is in ratio to the multitude of the different faces which can produce these numbers, and is as follows :