PROBABILITY being taken at random in AB, p shall fall in AX and q in XB. The whole number of cases is measured by m + n r 1 m + n /- l - = !(l-xydx->= =/ x*(l-x) n dx. |TO _J The number of favourable cases, when any particular set of p points, out of the p + q additional trials, falls in AX, is measured by m+n r because, the number of cases as to the m + n points being, when X falls on the element dx, | m + n ===-a;* l (l x)"dx , m
each of these affords x?(l - x) J cases where p new points fall on AX, and q on XB. Now, the number of different sets of p points being > I- / the required probability is Pq_ . . . . (15); or, by means of the known values of these definite integrals, p + q m+p
+ q
(16). For instance, the chance that in one more trial the event shall happen is ~. This is easy to verify, as the line AB has been ??i ~r ?i ~r Z divided into m + n + 2 sections by the m + n+l points taken on it (including X). Now if one more trial is made, i.e., one more point taken at random, it is equally likely to fall in any section ; and m + I sections are favourable. 19. When the number of trials m + n in art. 17 is large, the pro bability is considerable that the facility of the event on a single trial will not differ from its most natural value, viz., - by m + n more than a very small deviation. To make this apparent, v shall have to modify the formula (14), which gives for the chance that this facility lies between the limits a and $ (by substi tuting for the denominator its known value), (17). To find now the probability that the facility lies between the limits B = h 8 , and a = S , where 8 is small. Put for m + n m + n m x, - -fa;; and (In becomes m + n ~<S P +x S m + n J n m + n -x] dx . Now if x is small, and we put u = (a + x) m , X - ??1X logu^mlogct+m--- mx_mx* correct as far as the square of x. Hence the two factors under the sign of integration become mm ,(m+^)- ( -+ w l 2 ^ nn - ^- ( m +") 2 ^ im i and so that I TO + n + 1 m m ri (TO + n) n >5 n C ^( m + 7i ) + J c^n
- da . . (18).
Now, since by Stirling s theorem |-;/i = m m +*e- m V 2ir . the constant coefficient here becomes (m + n + l)(m + 7i)" t +"+ig-*-, v /2^ m m n n _ (m + nfi (19), m m . n n .c- m n 2ir^mn (m + n) m + n j taking m + n + l = m + n. Now if we substitute in (18) f2mn where or finally / -4-/N V/TT^^/O dt 775 (20), (21), for the approximate value of the probability that the real facility of the event lies between the limits - 8 1/1 + n Thus, if out of 10,000 trials, the event has happened 5000 times, the probability that, out of an infinite number, the number of successes shall lie between 4Tb, or between ^ and -/fe, of the whole, will be j>=-678 = f nearly, 10 6 for we find from (20) ^ = ^ 1Q4 / 2 ?b= ? nearly; and, referring to the table in art. 9, we find the above value for the integral (21). We must refer to the sixth chapter of Laplace for the investigation of how far the number of successes in a given number of fresh trials may be expected to deviate from the natural proportion, viz., that of the observed cases as also for several closely allied questions, with important applications to statistics. III. Ox EXPECTATION. 20. The value of a given chance of obtaining a given sum of money is the chance multiplied by that sum ; for in a great number of trials this would give the sum actually realized. The same may be said as to loss. Thus if it is 2 to 1 that a horse will win a race, it is considered a fair wager to lay 10 to 20 on the result ; for the value of the expected gain is | of 10, and that of the expected loss J of 20, which are equal. Thus, if the probabilities for and against an event are p, q, and I arrange in any way to gain a sum a if it happens and lose a sum b if it fails, then if pa^qb I shall neither gain nor lose in the long run ; but if the ratio a : b be less than this, my expectation of loss exceeds that of gain ; or, in other words, I must lose in the long run. The above definition is what is called the mathematical expecta tion ; but it clearly is not a proper measure of the advantage or loss to the individual ; for a poor man would undoubtedly prefer 500 down to the chance of 1000 if a certain coin turns up head. The importance of a sum of money to an individual, or its moral value, as it has been called, depends on many circumstances which it is impossible to take into account ; but, roughly and generally, there is no doubt that Daniel Bernoulli s hypothesis, viz., that this importance is measured by the sum divided by the fortune of the individual l is a true and natural one. Thus, generally speaking, 5 is the same to a man with 1000 as 50 to one with 10,000 ; and it may be observed that this principle is very generally acted on, in taxation, &c. 21. To estimate, according to this hypothesis, the advantage or moral value of his whole fortune to the individual, or his moral fortune, as Laplace calls it, in contradistinction to his physical fortune, letx = his physical fortune, i/ = his moral fortune, then, if the former receive an increment dx, we have, from Daniel Ber noulli s principle, y-k log T (22), k, h being two constants, x and y are always positive, and x>h ; for every man must possess some fortune, or its equivalent, in order to live. 22. To estimate now the value of a moral expectation. Suppose a person whose fortune is a to have the chance p of obtaining a sum a, q of obtaining ft, r of obtaining 7, &c. , and let p + q + r+ . . . =1, only one of the events being possible. Now his moral expectation from the first chance that is, the increment of his moral fortune into the chance is a + a , a ) 7 , . . , , g = lSir ( =pKiog(a + a)-pKiog( . Hence his whole moral expectation is 2 1 This rule must be understood to hold only when the sum is very small, or rather infinitesimal, strictly speaking. It would lead to absurdities if it were used for large increments (though Buffon has done so ; see Todhunter, p. 345). Thus, to a man possessing 100, it is of the sume importance to receive a gift of 100 as two separate gifts of 30 ; but this rule would give as the measure of the importance of the first |8R=1; while in the other case, it would give $&+ jiyiv^f . The real measure of the importance of an increment when not small is a matter for calculation, as shown in the text. 2 It is important to remark that we should be wrong in thus adding the expectations if the events were not mutually exclusive. For the mathematical
expectations it is not so.