PROBABILITY 773 12. We will give here a graphical representation (fig. 1), taken from M. Quetelet s Lettres sur la Theorie des Probability s, of the facilities of the different numbers of successes which may occur in 1000 trials as to any event which is equally likely to happen as not in each trial, as in 1000 tosses of a coin, or 1000 drawings from an urn containing one white and one black ball, replacing the ball each time, or again in drawing 1000 balls together from an urn containing a groat number of black and white in equal proportion. Asp = q = ^, we find from formula (8) that the chance of exactly half the entire number drawn, viz. , 500, being white is T= I == 02523; V5007T and the chance for any number 500 5 is found by multiplying _ s2 T by e~500. If then we take the central ordinate to represent T on any scale, and arrange along the horizontal line AB the different num bers of white balls which may occur, and erect opposite each number an ordinate representing the probability of that number, we have a graphical diagram of the relative possibilities of all possible proportions of black and white in the result. We see from it that all values of the number of white balls drawn less than 450, or greater than 550, may be considered impossible, the probabilities for them being excessively small. the JN white from A, or of the N white from B. As it is equally likely to have been any one of these, the chance that it came from A is N -f f N, or |. Suppose there had been two urns A and three urns B, and a white ball has been drawn from one of the five ; as in a great number N of drawings f N come from A and are white, |N from B and { of them are white, the chance that it came from one of the urns A is fHf+H)-f. In general suppose an event to have occurred which must have been preceded by one of several caitses, and let the antecedent probabilities of the causes be P 1 ,P 2 ,P 3 ... and let p^ be the probability that when the first cause exists the event will follow, p. 2 the same probability when the second cause exists, and so on, to find, after the event has occurred, the pro babilities of the several causes or hypotheses. Let a great number N of trials be made ; out of these the number in which the first cause exists is PjN, and out of this number the cases in which the event follows are ^PjN ; in like manner the cases in which the second cause exists and the event follows are /> 2 P 2 N ; and so on. As the event has happened, the actual case is one out of the number and as the number in which the first cause was present is j^PjN the a posteriori pro bability of that cause is ""i = , P _L ~ T> _L P j_ IT* (12). 500 Fig. 1. 1,80 The probability of the number of white balls falling between any two assigned limits, as 490 and 520, is found by measuring the area of the figure comprised between the two ordinates opposite those numbers, and dividing the result by the total area. II. PROBABILITY or FUTURE EVENTS DEDUCED FROM EXPERIENCE. 13. In our ignorance of the causes which influence future events, the cases are rare in which we know a priori the chance, or "facility," of the occurrence of any given event, as we do, for instance, that of a coin turning up head when tossed. In other cases we have to judge of the chances of it happening from experience alone. We could not say what is the chance that snow will fall in the month of March next from our knowledge of meteorology, but have to go back to the recorded facts. In walking down a certain street at 5 o clock on three different days, I have twice met a certain individual, and wish to estimate from these data the likelihood of again meeting him under the same circumstances in ignorance of the real state of things, viz., that he lives in that street, and returns from his business at that hour. Such is nearly the position in which we stand as to the probabilities of the future in the majority of cases. We have to judge then, from certain recorded facts, of the pro bability of the causes which have occasioned them, and thence to deduce the probabilities of future events occurring under the operation of the same causes. The term "cause " is not here used in its metaphysical sense, but as simply equivalent to "antecedent state of things." Let us suppose two urns, A containing two white balls, B con taining one white and one black ball, and that a person not know ing which is which has drawn a white ball from one, to find the probability that this is the urn A. This is in fact to find, suppos ing a great number of such drawings to be made, what proportion of them have come from the urn A. If a great number N of drawings are made indiscriminately from both urns, JN" come from the urn A and are all white, J N white come from the urn B, and N black. The drawing actually made is either one of So likewise for the other causes, the sum of these a posteriori probabilities being ir l + T 2 + ""3 + = 1 Supposing the event to have occurred as above, we now see how the probability as to the future, viz., whether the event will happen or fail in a fresh trial, is affected by it. If the first cause exists, the chance that it will happen is p 1 ; hence the chance of its happening from the first cause is p^i , so likewise for the second, third, &c. Hence the probability of succeeding on a second trial is P 1 v 1 + p 2 ir., + p. i v 3 + . . (13). 14. To give a simple example: suppose an urn to contain three balls which are white or black ; one is drawn and found to be white. It is replaced in the urn and a fresh drawing made ; find the chance that the ball drawn is white. There are three hypo theses, which are taken to be equally probable a priori, viz., the urn contains three white, two white, or one white, that of none white being now impossible. The probability after the event of the first is by (12) 4 + H + H > that of the second is , that of the third . Hence the chance of the new drawing giving a white ball is i+W+W-t- 15. The calculations required in the application of formulas (12) and (13) are often tedious, and such questions may often be solved in a simpler manner. Let us consider the following : An urn contains n black or white balls. A ball is drawn and replaced ; if this has been done r times, and in every case a white ball has appeared, to find the chance that the (r + l)th drawing will give a white ball. If s drawings are made successively from an urn containing n balls, always replacing the ball drawn, the number of different ways this may be done is clearly n . If there be n + 1 such urns, one with white balls, one with 1 white, one with 2 white, &c., the last with n white, the whole number of ways in which r drawings can be made from any one of them is (n+l)n r . Now the number of ways in which r drawings, all white, can be made from the first is 0, from the second 1, from the third 2 r , from the fourth 3 r , and so on ; so that the whole number of ways in which r drawings of a white ball can be made from the n + 1 urns is l r + 2 r +3 r + . . . n r . Hence the chance that if r drawings are made from an urn containing
n black or white balls all shall be white is