THE DOCTRINE OF CHANCES. 15
From what has been said it will be obvious that although we may with an equality of chance contend about the happening of an event once in a certain number of trials, vet wTe cannot contend for its happening twice in a double number of trials, or three times in triple that number; and so on. Thus, although the chances are equal of throwing two aces with two dice in 25 throws, yet we cannot undertake that the two aces shall come up twice in 50 throws, the number requisite being 58 or 59 times; and much less, that it will come up three times in 75 throws, the number requisite being 93 and 94 ; so that we cannot undertake that in a very great number of trials the happening shall be oftener than in the proportion of 1 to 36. And therefore we may lay it down as a maxim, that events at long run will not happen oftener than in the proportion of the chances they have to happen in one trial, and if we assign any other proportion, the odds against us will increase continually.
Analysis of the Chances, or the Points produced by two or more Dice.
With two dice it is evident that we may produce thirty-six different combinations, for each of the six faces of one may be successively combined six times with each of the six faces of the other. Therefore, with a number of dice = n, the number of different combinations they will produce will be 6n
The odds of throwing doublets, therefore, with two dice, are 35 to 1.