142 DE MOIVRE. 



chosen Examples, but tliat be bad added to it several curious things of 

 bis own Invention. 



Since the printing of my Specimen, Mr. de Monmort, Author of the 

 Analyse des jeux de Hazard^ Published a Second Edition of that Book, 

 in which he has particularly given many proofs of his singular Genius, 

 and extraordinary Capacity; which Testimony I give both to Truth, 

 and to the Friendship with which he is pleased to Honour me. 



The concluding paragraph of the preface to the first edition 

 refers to the Ars Conjectandi, and invites Nicolas and John Ber- 

 noulli to prosecute the subject begun in its fourth part ; this 

 paragraph is omitted in the other editions. 



We repeat that we are about to analyse the third edition of the 

 Doctrine of Chances, only noticing the previous editions in cases of 

 changes or additions in matters of importance. 



257. The Doctrine of Chances begins with an Introduction of 

 S3 pages, which explains the chief rules of the subject and illus- 

 trates them by examples ; this part of the work is very much fuller 

 than the corresponding part of the first edition, so that our remarks 

 on the Introduction do not apply to the first edition. De Moivre 

 considers carefully the following fundamental theorem : suppose 

 that the odds for the happening of an event at a single trial are as 

 a to h, then the chance that the event will happen r times at least 

 in n trials is found by taking the first n — r-i-1 terms of the expan- 

 sion of (a + hy and dividing by (a + by. We know that the result 

 can also be expressed in another manner corresponding to the 

 second formula in Art. 172 ; it is curious that De Moivre gives 

 this without demonstration, though it seems less obvious than 

 that which he has demonstrated. 



To find the chance that an event may happen just r times, De 

 Moivre directs us to subtract the chance that it will happen at least 

 r—1 times from the chance that it will happen at least r times. 

 He notices, but less distinctly than we might expect, the modern 

 method which seems more simple and more direct, by which we 

 begin with finding the chance that an event shall happen jvst r 

 times and deduce the chance that it shall happen at least r 

 times. 



