ON LIFE CONTINGENCIES. 159 



the question of the truth or falsehood of the preceding 

 notion, it will be necessary to ask what laws the 

 duration of human life follows, and whether it fol- 

 low any laws at all ? Take two separate hundreds 

 of persons, each aged twenty, is there any reason to 

 conclude that the united lives of all the first hundred 

 will make an amount of years nearly equal to that of 

 the second ? 



In order to try this point, I shall take another question, 

 yet more unfavourable to the result which I wish 

 to establish. In 100 persons all aged twenty, we know 

 that there is but a very slight chance that any given 

 one of them shall reach the age of eighty; and we 

 may consider it a certainty (or of an extremely high 

 probability), that none of them will see the age of a 

 hundred and twenty. We will consider it therefore as 

 given, that no one shall live to the last-mentioned age, 

 and we will even suppose that all ages of death between 

 20 and 120 are equally probable. This of course 

 very much increases our chance of fluctuation : but 

 even with this supposition it is not very great. 



Let us suppose a lottery in which there are counters 

 marked with every possible number or fraction inter- 

 mediate between and E : so that the drawing may 

 have any mark whatsoever. If then we draw out 100 

 counters, the least possible amount of drawings will be 0, 

 the greatest 100 times E : and if all drawings be 

 equally probable, we have no reason to suppose that 

 our amount will exceed 50 times E, which does not 

 equally apply in favour of its falling short of that 

 quantity. That we shall have exactly 50 times E, is 

 an event of which the chance is infinitely small : but 

 that the amount shall lie between limits which are 

 tolerably near 50 times E, is very probable. 



PROBLEM. Let there be counters, in equal numbers, 

 with every possible mark between and E. What is the 

 probability that the average of n drawings shall not differ 

 from the half of E, one way or the other, by more than Jc. 



RULE. Multiply k by the square root of six times 



