The maximum value (see CALCULUS, halm TFSIMAL, Maxima and Minima) of rs (1 — r)*-8 is obtained by putting r=— and is •(n—s)/1-6 • Then the most probable cause is that in which the actual ratio of the number of white balls to the total number is the same as the corre sponding ratio in the observed results.
A discussion of the value of P, which must be omitted, yields the following notable result : Theorem 5. The probability that, under the conditions of example 9, the ratio of the number of white balls to the total number shall not deviate from its most probable value, , by y s is 2 more than y V o Empirical Probabilities.— The preceding theory finds application in connection with any sets of events which for the purposes of any discussion may justly be compared with the drawing of balls from a bag. In some of the important connections this theory is in fact indispensable. It will be noted that in both of the examples 8 and 9 the inquiry was in regard to the probability of the various possible num bers of white and black balls. But if the ratio of the number of white balls to the total num ber of balls be known, that ratio furnishes the probability that a further single drawing will produce a white ball; and in general this proba bility is what is most urgently sought. Proba bilities estimated in this way from observed events are called empirical probabilities.
Two methods of deriving empirical proba bilities are used, one theoretically correct, but leading to troublesome computations, the other avowedly approximate only. The latter method
involves the assumption that, of the possible causes of the observed event, that which has the greatest a posteriori probability is the true one. These two methods will be applied to example 8.
Suppose that after the drawing of the balls from the bag the question be asked, "What now is the probability of drawing a white ball?" By the second method the answer would be obtained as follows: The most probable state of the bag is that it contains one black and four white balls. Hence the probability of drawing a white ball is I.
The following is the other method of treat ment: The probability that the bag contains four white balls and one black one and that a white one will be drawn from it is X /: that it contains three white and two black and that a white one will be drawn is X I; that it contains two white and three black and that a white one will be drawn is X ; and that it contains one white and four black, and that a white one will be drawn is A, X I. Hence the probability that a white ball will be drawn is H.
Where the number of causes is greater the difference between the two values is smaller, and theorem 5 shows that when the number is very great the difference can be very small indeed, as it becomes nearly certain that the ratio r differs by very little from