Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/39

Rh 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