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.
The inductive part of the problem is completed, since we have found that the urn most likely contains three white and one black ball, and have assigned the exact probability of each possible supposition. But we are now in a position to resume deductive reasoning, and infer the probability that the next drawing will yield, say a white ball. For if the box contains three white and one black ball, the probability of drawing a white one is certainly 3/4; and as the probability of the box being so constituted is 27/46, the compound probability that the box will be so filled and will give a white ball at the next trial, is
27/46 × 3/4 or 81/184.
Again, the probability is 16/46 that the box contains two white and two black, and under those conditions the probability is 1/2 that a white ball will appear; hence the probability that a white ball will appear in consequence of that condition, is
16/46 × 1/2 or 32/184.
From the third supposition we get in like manner the probability
3/46 × 1/4 or 3/184.
Since one and not more than one hypothesis can be true, we may add together these separate probabilities, and we find that
81/184 + 32/184 + 3/184 or 116/184