4. If either A or n must happen, a + b =1 ; and if n trials are to be made, the several terms in the expansion of (a are the chances of the arrivals denoted by their exponents. Thus a" ie the chance that all are As.
a that n-1 are AZ and one is B.
n n1 that n-2 are As and two are Bs, &c. 2 5. When an event has happened which may have sprung from any one of the sets of circumstances which we may describe by A, a, c, &c., and it is desired to find what is the probability of the event which did happen having arisen from one or another set, proceed as follows : -Had the circumstances denoted by A certainly existed, and the event been a contingency, let a be the probability that those circum stances would have produced the event ; or briefly, let a be the pro bability that A, when brings about the event. Let b be the same for a certain, c for c, dre. Then the probability that it was A which brought about the event which did happen, is the first of the following set : a a+b+c+&c., a+b+e+&c., a+b+c+&c.' the probability that it was a is the second, that it was c the third, and so on. Or, when convenient, instead of a, b, c, &c., may be substituted attach to our conclusions. That person was Dr. Wollaston, who, upon a given point, was induced to offer me a wager of two to one ou the affirmative. I rather Impertinently quoted Boner's well-known lines ' Quoth Abe, I've heard old cunning stagers Say fools for arguments use wagers,' about the kind of persons who use wagers for argument, and he gently explained to me that he considered such a wager not as a thoughtless thing, hut as the expression of the amount of belief in the mind of the person offering It ; com bining this curious application of the wager, as a meter, with the necessity that ever existed of drawing conclusions, Lot absolute, but proportionate to the evidence." any whole numbers which are proportional to them. For example, let there be three lotteries, containing balls as follows : all white, 4 white, 1 black, I I 2 white, 7 black. I A drawing has been made three times from one of these, but from which is not known, the ball being replaced after the drawing, and every drawing has given a white ball. Required the chances in favour of each of these lotteries having been the one drawn from. If it had
been the first, the chance of the event would have been 1 (or certainty) ; if it had been the second, the chance of the event would have been 4 4 4 64 x x or, 125' if it had been the third, the chance would have 2 2 2 8 been - 9 9 9" 729' x - x - — The numerators of these fractions, reduced to a common denominator, are 91125, 46656, and 1000; whence the probability that the lottery drawn from was the first, is 91125 91125 91125+46656+1000' or 138781' and the probabilities of the second and third lotteries are 46656 1000 138781 and 138781' The preceding question is well calculated to show the meaning of questions in this theory, which is thus seen to be applied to events, not because they are uncertain, but because they are unknown. So soon as the lottery is chosen, it is certain which it is, but since it is not known to the drawer, it is to him as much a contingency as it was before it was chosen.
6. When, in such a case as the preceding, it is required to know the probabilities of the events which may happen at any further trials, the probability of each lottery having been the one in question is to be multiplied by the probability of the new event arising from that lottery taken as certain, and the results added together. Thus, suppose there are two lotteries, one having all white balls, and the other equal numbers of white and black balls ; two drawings have been mado from one of them (not known) and both drawings have given a white ball; what is the probability that a third drawing from the same lottery would also give a white ball ? The chances for the two lotteries are found 4 1 by the last to be 3 and while the chances of a white ball from 1 4 1 one and the other are I and -1. It follows that x x or 9 10' is the chance of the third drawing giving a white ball.
7. If A or n must happen at every trial, and if in m trials nothing but A has happened, and if we know nothing whatever about the nature of preceding circumstances, then it is ra + 1 to 1 that A shall happen at the next trial, and m+1 to k that A shall happen throughout the next k trials. But if in + n trials A have occurred m times and B n times, it is m+1 to n+1 that A shall occur at the next trial.