774 PROBABILITY for all we know of the contents of such an urn is that they are equally likely to be those of any one of the n + l urns above. If now a great number N of trials of r drawings be made from such uriis, the number of cases where all are white is p r N . If r+ 1 drawings are made, the number of cases where all are white is p r+1 X ; that is, out of the p r N cases where the first r drawings are white there are^v+iN where the (r + l)th is alsojwhite ; so that the probability sought for in the question is p r+} 1 iH-i + 2--+i + 3 r + 1 + . . . n r + l " p r ~n l r + 2 r + 3 r + . . . n r 16. Let us consider the same question when the ball is not replaced. First suppose the n balls arranged in a row from A to B as below, the white on the left, the black on the right, the arrow marking the point of separation, which point is unknown (as it would be to a blind man), and is equally likely to be in any of its n + 1 possible positions. A 1 2 B OOOOOOOOOOO. ? Now if two balls, 1 and 2, are selected at random, the chance that both are white is the chance of the arrow falling in the divi sion 2B of the row. But this chance is the same as that of a third ball 3 (different from 1 and 2), chosen at random, falling in 2B, which chance is J, because it is equally probable that 1, 2, or 3 shall be the last in order. It is easy to see that these chances are the same if we reflect that, the ball 3 being equally likely to fall in Al, 12, or 2B, the number of possible positions for the arrow in each division always exceeds by 1 the number of positions for 3 ; therefore as 3 is equally likely to fall in any of the three divi sions, so is the arrow. The chance that two balls drawn at random shall both be white is thus ^ ; in the same way that for three balls is J, and so on. Hence the chance that r balls drawn shall all be white is the same chance for r + 1 balls is thus, as in a large number N of trials the number of cases where the first r drawn are white is p r ^, and the number where the first r + 1 are white is p r+l N, we have the result : If r balls are drawn and all prove to be white, the chance that the next drawn shall also be white is ^p r +i = r + I p r r + 2 This result is thus independent of n, the whole number of balls. This result applies to repeated trials as to any event, provided we have really no a priori knowledge as to the chance of success or failure on one trial, so that all values for this chance are equally likely before the trial or trials. Thus, if we see a stranger hit a mark four times running, the chance he does so again is ; or, if a person, knowing nothing of the water where he is fishing, draws up a fish each time in four casts of his line, the same is the chance of his succeeding a fifth time. 1 In cases where we know, or rather think we know, the facility as to a single trial, if the result of a number of trials gives a large ditference in the proportion of successes to failures from what we should anticipate, this will afford an appreciable presumption that our assumption as to the facility was erroneous, as indeed common sense indicates. If a coin turns up head twenty times running, we should say the two faces are probably not alike, or that it was not thrown fairly. We shall see later on, when we come to treat of the combination of separate probabilities as to the same event, the method of dealing with such cases (see art. 39). We will give another example which may be easily solved by means of (12), or by the simpler process below. There are n horses in a race, about which I have no knowledge eicept that one of the horses A is black ; as to the result of the race I have only the information that a black horse has certainly won : to find the chance that this was A supposing the propor tion of black among racehorses in general to be p ; i.e., the pro bability that any given horse is black is p. Suppose a large number N of trials made as to such a case. A wins in N of these. Another horse B wins in N ; out of these _ n _ n 1 It may be asked why the above reasoning does not apply to the case of ie chance of a coin which has turned up head r times doing so once more. The reason Is that the antecedent probabilities of the different hypotheses are not equal. Thus, let a shilling have turned up head once; to find the chance of its doing so a second time. In formula (12) three hypotheses may be made as to a double : throw 1 two heads, 2 a head and tail, 3 two tails ; but> the proba- es of these are respectively J, i, J; therefore by (12) the probability of the r SS eTent l3 H-(i+l D=J : tnat of the second is also J; and by (13) tne probability of succeeding on a second trial is because, if hypothesis 2 is the true one, the second trial must fail. B is black in NT>. Likewise for C : and so on. Hence the actual n case which has occurred is one out of the number 1 n - ! JS H Nw : n n and, as of these the cases in which A wins are -N, the required chance that A has won is 1 .- 17. We now proceed to consider the important theorem of Bayes (see Todhunter, p. 294 ; Laplace, Theorie Analytique dcs Prob., chap. 6), the object of which is to deduce from the experience of a given number of trials, as to an event which must happen or fail on each trial, the information thus afforded as to the real facility of the event in any one trial, which facility is identical with the proportion of successes out of an infinite number of trials, were it possible to make them. Thus we find in the Carlisle Table of Mortality that of 5642 persons aged thirty 1245 died before reaching fifty ; it becomes then a question how far we can rely on the real facility of the event, that is, the proportion of mankind aged thirty who die before fifty not differing from the ratio il|f by more than given limits of excess or defect. Again, it may be asked, if 5642 (or any other number of) fresh trials be made, what is the probability that the number of deaths shall not differ from 1245 by more than a given deviation ? The question is equivalent to the following : An urn contains a very great number of black and white balls, the proportion of each being unknown ; if, on drawing m + n balls, m are found white and n black, to find the probability that the proportion of the numbers in the urn of each colour lies between given limits. The question will not be altered if we suppose all the balls ranged in a line AB (fig. 2), the white ones on the left, the black on the right, the point X where they meet being unknown and all positions for it in AB being a priori equally probable. Then, m + n points having r been chosen at random in AB, m are found to - fall on AX, n on XB. That is, all we know of X is that it is the (i + l)th in order beginning from A of m + n + l points chosen at random in AB. If we put AB = 1, AX = x, the number of cases when the point X falls on the element dx, is measured by since for a specified set of m points, out of the m + n, falling on AX, the measure would be x m (l - x} n dx, and the number of such sets is j j . Now the whole number of cases is given by integrat- [ 1)1 I ?t ing this differential from 1 to ; and the number in which X falls between given distances o, /3 from A is found by integrating from /3 to o. Hence the probability that the ratio of the white balls in the urn to the whole number lies between any two given limits a, & is P- //3 x* (l - x) n dx 1 X m (l - x)"dx (14). The curve of frequency for the point X after the event that is, the ordiuate of which at any point of AB is proportional to the fre quency or density of the positions of X in the immediate vicinity of that point is 2/ = a: m (l -x)" ; the maximum ordinate KV occurs at a point K, dividing AB in the ratio TO : n, the ratio of the total numbers of white and black balls being thus more likely to be that of the numbers of each actually drawn than any other. Let us suppose, for instance, that three white and two black have been drawn ; to find the chance that the proportion of white balls is between f and of the whole ; that is, that it differs by less than ^ from $, its most natural value. P f* = f r ~ I xl- x)"dx Jo 2256 18 . = _=- nearly. 18. An event has happened m times and failed n times in m + n trials. To find the probability that in p + q further trials it shall
happen ^> times and fail q times, that is, that, p + q more points