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12 THE DOCTRINE OF CHANCES. 







From the foregoing problems it appears that, when A wants but one game of a set, and B two, the odds in favour of the former are 3 to 1. The accuracy of this calculation, however, has been questioned by the celebrated d'Alembert, who illustrates his position by the game of Croix ou Pile (Heads or Tail), which is too well known to need a definition. 







In these four combinations there is only one by which the thrower loses; the odds are then 3 to 1 in his favour. If he betted in three coups, he would find eight combinations, seven in his favour, and one against him ; the odds would be, therefore, 7 to 1 ; but, says d'Alembert, is this correct ? For to consider only the two coups, must we not reduce to one the two combinations, 
