NICOLAS BERNOULLI. 195 



value of an annuity on a life ; the sum to be paid to assure to a 

 child just born an assigned sum on his attaining a certain age ; 

 marine assurances ; and a lottery problem. He also touches on the 

 probability of testimony ; and on the probability of the innocence 

 of an accused person. 



The essay does not give occasion for the display of that mathe- 

 matical power which its author possessed, and which we have seen 

 was called forth in his correspondence with Montmort ; but it indi- 

 cates boldness, originality, and strong faith in the value and extent 

 of the applications which might be made of the Theory of Pro- 

 bability. 



We will take two examples from the Essay. 



34:0. Suppose there are h men who will all die within a years, 

 and are equally likely to die at any instant ^\dthin this time : re- 

 quired the probable duration of the life of the last survivor. 

 Nicolas Bernoulli really views the problem as equivalent to the 

 following : A line of length a is measured from a fixed origin ; on 

 this line h points are taken at random : determine the mean dis- 

 tance from the origin of the most distant point. 



Let the line a be supposed divided into an indefinitely large 

 number n of equal parts ; let each part be equal to c, so that 

 nc = a. 



Suppose that each of the h points may be at the distance 

 c, or 2c, or Sc, ...up to nc\ but no two or more at exactly the 

 same distance. 



Then the whole number of cases will be the number of combi- 

 nations of n things taken 5 at a time, say </> {n, h). 



Suppose that the most distant point is at the distance xc ; then 

 the number of ways in which this can happen is the number of 

 ways in which the remaining ^ — 1 points can be put nearer to the 

 origin ; that is, the number of combinations of a; — 1 things, taken 

 J — 1 at a time, say </> (a? — 1, h — 1). 



Hence the required mean distance is 



^ xc (j) {x — \, h — V) 



where the summation extends from x = h to x = n. 



13—2 



