184 I>E MOIVRE. 



...I'll take the liberty to say, tliat tliis is tlie hardest Problem that 

 can be projDosed on the Subject of Chance, for which reason I have re- 

 served it for the last, but I hope to be forgiven if my Solution is not 

 fitted to the capacity of all Headers; however I shall derive from it 

 some Conclusions that may be of use to every body : in order thereto, 

 I shall here translate a Paper of mine which was printed November 12, 

 1733, and communicated to some Friends, but never yet made public, 

 reserving to myself the right of enlarging my own Thoughts, as occasion 

 shall require. 



Then follows a section entitled A Method of a2yproximating the 

 Sum of the Terms of the Bmomial (a -1- b)" expanded into a Series, 

 from lohence are deduced some practical Rules to estimate the 

 Degree of Assent which is to he given to Experiments. This section 

 occupies pages 243 — 254 of the Doctrine of Chances; we shall find 

 it convenient to postpone our notice of it until we examine the 

 Miscellanea Analytica. 



325. De Moivre's Problem LXXIV. is thus enunciated : 



To find the Probability of throwing a Chance assigned a given 

 ' number of times without intermission, in any given number of Trials. 



It was introduced in the second edition, page 243, in the fol- 

 lowing terms : 



When I was just concluding this Work, the following Problem was 

 mentioned to me as very difficult, for which reason I have considered it 

 with a particular attention. 



De Moivre does not demonstrate his results for this problem ; 

 we will solve the problem in the modern way. 



Let a denote the chance for the event in a single trial, h the 

 chance against it ; let n be the number of trials, p the nvimber of 

 times without intermission for which the event is required to hap- 

 pen. We shall speak of this as a run of p. 



Let Un denote the probability of having the required run of ^? 

 in n trials ; then 



«^,i+i = y-n + (1 - '^fn-j^ 'bdF : 



for in n + 1 trials we have all the favourable cases which we have 

 in n trials, and some more, namely those in which after having 

 failed in n—p trials, we fail in the (n— ^? + l)"' trial, and then 

 have a run of p. 



