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18 THE DOCTRINE OF CHANCES. 



If, therefore, we bet to throw 11 the first time with two dice, the odds are 2 to 34, and if 7, 6 to 30, there being six ways by which 7 may be thrown, and thirty against it. We must, however, observe that in the eleven different numbers which may be thrown with two dice, 7, which is the mean proportional between 2 and 12, has more chances than the others, which on their side have more or less chances in their favour as they approach the two extremes.
This difference of the multitude of chances produced by the mean numbers compared to the extreme, increases considerably in ratio to the number of dice. It is such, that if we make use of seven dice, which produce points from 7 up to 42, we shall find that we shall almost invariably throw the mean numbers 24 and 25, or those which approach the nearest to them, viz. 22, 23, 26, 27 ; and if, instead of seven dice, we make use of twentyfive, which will produce numbers from 25 to 150; we might with safety bet an equal wager to throw 86 and 87
The above remaik is important, as it must tend to expose at a glance the gross imposition of those 
