77(5 PROBABILITY and, if Y stands for his moral fortune including this expectation, that is, k log ~ + E, we have
fcp}^g(a + a) + kq g(a + /J)+ . . . - klogh . . (23).
Let X be the physical fortune corresponding to this moral one, by (22) Y = k log X - k log h . Hence X.-=(a + a)> (a + fi)i(a + y) r (24); and X a will be the actual or physical increase of fortune which is of the same value to him as his expectation, and which he may reasonably accept in lieu of it. The mathematical value of the same expectation is 2)a + q0 + ry+ (25). 23. Several results follow from (24). Thus, if the sums o, , y . . . are very small, it is easy to see that the moral expectation coincides with the mathematical, for 24. We may show also that it is disadvantageous to play at even a fair game of chance (unless the stakes are very small, in which case the last article applies). Thus, suppose a man whose fortune is a plays at a game where his chance of winning a sum o is p, and his chance of losing a sum /} is q = l -p. If the game is fair, Now by (24) the physical fortune which is equivalent to his prospects after the game is Now the geometrical mean of r quantities is less than the arith metical, 1 so that if there are /3 quantities a + a, and a quantities o-ft , &r o a ) a+ 3 ^3( + a) + a(a - ) (a + a) (a - 0) > <. or X<a, so that he must expect morally to lose by the game. 25. The advantage of insurance against risks may be seen by the following instance. A merchant, whose fortune is represented by 1, will realize a sum e if a certain vessel arrives safely. Let the probability of this be^;. To make up exactly for the risk run by the insurance company, he should pay them a sum (!-;>). If he does, his moral fortune becomes by (22) h while, if he does not insure, it will be (23). T 1 kplog . Now the first of these exceeds the second, so that he gains by insuring on these terms ; because that is , 771 for, putting w = , fo ^ m+n m + n because (see note art. 24), if m (1 + ) + n is divided into m + n equal parts, their product is greater than that of m parts each equal to 1 + ( and n parts each equal to 1. The merchant will still gain by paying, over and above what covers the risk of the company, a sum a, at most, which satisfies log (1 - a+pt) =.plog (1 + ) ; By paying any sum not exceeding this value, he still gains, while 1 A very simple proof of this principle is as follows : let, a number N be divided into r parts a, b, c, &c.; if any two of these, as a, b, are unequal, since it follows that the product abed ... is increased by substituting ^J- , -^ , for a and b. Hence as long as any two are unequal we can divide N differently so as to obtain a greater product; and therefore when the parts we all equal the product is gre itest, or /a + b+c+ . . .r^ the insurance office also makes a profit, which is really a certainty when it has a large business ; so that, as Laplace remarks, this example explains how such an office renders a real service to the public, while making a profit for itself. In this it differs from a gambling establishment, in which case the public must lose, in any sense of the term. It may be shown that it is better to expose one s fortune in separate sums to risks independent of each other than to expose the whole to the same danger. 2 Suppose a merchant, having a for tune a, has besides a sum e which he must receive if a ship arrives in safety. By (24) the value in money of his present fortune is X = (a + f )rai, where ^> = chance of the ship arriving, and q=I -p. Now suppose he risks the same sum in two equal portions, in two ships. We cannot apply (23), as the events are not mutually exclusive ; but we see that, if both ships arrive, the chance of this being p 1 , he realizes the whole sum e ; if one only arrives, the chance being 2pq, he receives |e ; if both are lost, the chance being q", he loses all. Thus (24) he is now worth a sum Now this sum is greater than the former ; for (a + e )P -P. (a + ^-M. a?-* > 1 , that is, (a + f)-^(a + Jf)*W for (jLi?); as is obviously true. Now suppose he risks the sum e in three separate ventures. His fortune will be . (a + e ) 8 ^. (a + l^i ; and we have to show that this is worth more than when there were two. If we put a outside each bracket, and put 8 = 5- we have to oct prove (1 + 35)^(1 + 25)"^. (1 + 8) 3 M 2 > (1 + 35) J. (1 + f 5) 2 ; 18) 2 , or, since p* - p = - pq, S)-*(1 + (1 + 28)3 hence the fraction in the brackets is always less than its _pth power as p < 1 ; and we can now show that that is, (l + 25) 3 >(l + 35)(l+fS) 2 , Laplace shows (ch. x.) that the gain continues to increase by subdivision of the risk ; it could no doubt be shown by ordinary algebra. He shows further that the moral advantage tends to become equal to the mathematical. This may be done more easily thus : The expression is, when e is divided into r equal parts,
- - 1
rp g a + f- i 2 and we have to find the limit towards which this tends as r becomes infinitely great. Put 2 = a + e ; erpr-iq/ r ( r -^) p r -<i 2 / e rq r - l p -) ( 2 -2-r2 P . . .(z-(r-l)-} 12-c rj rj r Now in the binomial expansion the greatest term is the (<?r + l)th, viz., r(r-l) . . . (rp + l) r p rq 1-2-3 . . . rq p * The factor in X corresponding to this is ( f T TT T (z-rg-) -U, if we put U = 2 - qe . Let us now express the binomial series before and after T thus : 2 The familiar expression net to " put all one s eggs in the same basket " shows
us how general common sensj lias recognized this principle.