[167] *Traité élémentaire du Calcul des Probabilités*, 3rd ed. (1833), p. 148. *Simple Illustration of the Inverse Problem.* Suppose it to be known that a ballot-box contains only four black or white balls, the ratio of black and white balls being unknown. Four drawings having been made with replacement, and a white ball having appeared on each occasion but one, it is required to determine the probability that a white ball will appear next time. Now the hypotheses which can be made as to the contents of the urn are very limited in number, and are at most the following five:-- 4 white and 0 black balls 3 " " 1 " " 2 " " 2 " " 1 " " 3 " " 0 " " 4 " " The actual occurrence of black and white balls in the drawings puts the first and last hypothesis out of the question, so that we have only three left to consider. If the box contains three white and one black, the probability of drawing a white each time is 3/4, and a black 1/4; so that the compound event observed, namely, three white and one black, has the probability 3/4 × 3/4 × 3/4 × 1/4, by the rule already given (p. 204). But as it is indifferent in what order the balls are drawn, and the black ball might come first, second, third, or fourth, we must multiply by four, to obtain the probability of three white and one black in any order, thus getting 27/64. Taking the next hypothesis of two white and two black balls in the urn, we obtain for the same probability the quantity 1/2 × 1/2 × 1/2 × 1/2 × 4, or 16/64, and from the third hypothesis of one white and three black we deduce likewise 1/4 × 1/4 × 1/4 × 3/4 × 4, or 3/64. According, then, as we adopt the first, second, or third hypothesis, the probability that the result actually noticed would follow is 27/64, 16/64, and 3/64. Now it is certain that one or other of these hypotheses must be the true one, and their absolute probabilities are proportional to the probabilities that the observed events would follow from them (pp. 242, 243). All we have to do, then, in order to obtain the absolute probability of each hypothesis, is to alter these fractions in a uniform ratio, so that their sum shall be unity, the expression of certainty. Now, since 27 + 16 + 3 = 46, this will be effected by dividing each fraction by 46, and multiplying by 64. Thus the probabilities of the first, second, and third hypotheses are respectively-- 27/46, 16/46, 3/46.