90 numbers is 4005, and that of the combinations two by two of 5 numbers is 10. The probability of the drawing of a given pair is then 1 ⁄ 4005 , and the lottery ought to give four hundred and a half times the stake; it ought to give 11748 times for a given tray, 511038 times for a quaternary, and 43949268 times for a quint. The lottery is far from giving the player these advantages.
Suppose in an urn a white balls, b black balls, and after having drawn a ball it is put back into the urn; the probability is asked that in n number of draws m white balls and n - m black balls will be drawn. It is clear that the number of cases that may occur at each drawing is a + b. Each case of the second drawing being able to combine with all the cases of the first, the number of possible cases in two drawings is the square of the binomial a + b. In the development of this square, the square of a expresses the number of cases in which a white ball is twice drawn, the double product of a by b expresses the number of cases in which a white ball and a black ball are drawn. Finally, the square of b expresses the number of cases in which two black balls are drawn. Continuing thus, we see generally that the nth power of the binomial a + b
expresses the number of all the cases possible in n draws; and that in the development of this power the term multiplied by the mth power of a expresses the number of cases in which m white balls and n - m black balls may be drawn. Dividing then this term by the entire power of the binomial, we shall have the probability of drawing m white balls and n - m black balls. The ratio of the numbers a and a + b being the probability of drawing one white ball at one draw; and the ratio of the numbers b and a + b being the probability of drawing one black ball; if we call these probabilities p and q, the probability of drawing m white balls in n draws will be the term multiplied by the mth power of p in the development of the nth power of the binomial p + q; we may see that the sum p + q is unity. This remarkable property of the binomial is very useful in the theory of probabilities. But the most general and direct method of resolving questions of probability consists in making them depend upon equations of differences. Comparing the successive conditions of the function which expresses the probability when we increase the variables by their respective differences, the proposed question often furnishes a very simple proportion between the conditions. This proportion is what is called equation of ordinary or partial differentials; ordinary when there is only one variable, partial when there are several. Let us consider some examples of this.
Three players of supposed equal ability play together on the following conditions: that one of the first two players who beats his adversary plays the third, and if he beats him the game is finished. If he is beaten, the
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