Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/40

30 victor plays against the second until one of the players has defeated consecutively the two others, which ends the game. The probability is demanded that the game will be finished in a certain number n of plays. Let us find the probability that it will end precisely at the nth play. For that the player who wins ought to enter the game at the play n — 1 and win it thus at the following play. But if in place of winning the play n — 1 he should be beaten by his adversary who had just beaten the other player, the game would end at this play. Thus the probability that one of the players will enter the game at the play n — 1 and will win it is equal to the probability that the game will end precisely with this play; and as this player ought to win the following play in order that the game may be finished at the nth play, the probability of this last case will be only one half of the preceding one. This probability is evidently a function of the number n; this function is then equal to the half of the same function when n is diminished by unity. This equality forms one of those equations called ordinary finite differential equations.

We may easily determine by its use the probability that the game will end precisely at a certain play. It is evident that the play cannot end sooner than at the second play; and for this it is necessary that that one of the first two players who has beaten his adversary should beat at the second play the third player; the probability that the game will end at this play is $1⁄2$. Hence by virtue of the preceding equation we conclude that the successive probabilities of the end of the game are $1⁄4$ for the third play, $1⁄8$ for the fourth play, and so