388 Insurance and Gambling. [CHAP. xv. 



genealogy, and various fanciful accounts have been given 

 by those unfamiliar with the theory of probability. What 

 is often apt to be overlooked is the extreme slightness of 

 what may be termed the &quot;turn of the tables&quot; in favour 

 of the survival at each generation. In the above numerical 

 example we have made an extravagantly favourable sup 

 position, by assuming that the population doubles at every 

 generation. In an old and thickly populated country where 

 the numbers increase very slowly, we should be much nearer 

 the mark in assuming that the average effective family, 

 that is, the average number of children who live to marry, 

 was only two. In this case every family which was repre 

 sented at any time by but a single male would have but 

 three chances in four of surviving extinction, and of course 

 the process of thinning out would be a more rapid one. 



17. The most interesting class of attempts to prove 

 the disadvantages of gambling appeal to what is tech 

 nically called moral expectation as distinguished from 

 mathematical expectation. The latter may be denned 

 simply as the average money value of the venture in 

 question; that is, it is the product of the amount to be 

 gained (or lost) and the chance of gaining (or losing) it. 

 For instance, if I bet four to one in sovereigns against the 

 occurrence of ace with a single die there would be, on the 

 average of many throws, a loss of four pounds against a gain 

 of five pounds on each set of six occurrences; i.e. there 

 would be an average gain of three shillings and fourpence 

 on each throw. This is called the true or mathematical 

 expectation. The so-called moral expectation , on the other 

 hand, is the subjective value of this mathematical expec 

 tation. That is, instead of reckoning a money fortune in 

 the ordinary way, as what it is, the attempt is made to 

 reckon it at what it is felt to be. The elements of compu- 



