1 00 MONTMORT. 



177. Montmort devotes his pages 248 — 257 to the discussion 

 of a game of Bowls, which leads to a problem resembling the Pro- 

 blem of Points. The problem was started by De Moivre in his 

 Be Mensura Sortis ; see Montmort, page 866, and the Doctrine of 

 Chances, page 121. De Moivre had supposed the players to be of 

 equal skill, and each to have the same number of balls ; Montmort 

 generalised the problem by supposing players of unequal skill and 

 having unequal numbers of balls. Thus the problem was not in 

 Montmort's first edition. 



Montmort gives on his page 256 a simple example of a solution 

 of a problem which appears very plausible, but which is incorrect. 

 Suppose A plays with one bowl and B with two bowls ; required 

 their respective chances in one trial, assuming equal skill. 

 Considering that any one of the three bowls is as likely as the 



.2 .1 



others to be first, the chance of ^ is ^r and that of ^ is - . But by 



3 3 -^ 



the incorrect solution Montmort arrives at a different result. For 



suppose A to have delivered his bowl. Then B has the chance 



^ with his first bowl of beating A ; and the chance - x ^ of failing 

 with his first bowl and being successful with his second. Thus ^'s 

 chance appears to be - • Montmort considers the error of this so- 

 lution to lie in the assumption that when B has failed to beat A 

 with his first bowl it is still an even chance that he will beat A with 

 his second bowl : for the fact that B failed with his first bowl 

 suggests that ^'s bowl has a position better than the average, so 

 that jB's chance of success with his second bowl becomes less than 

 an even chance. 



178. Montmort then takes four problems in succession of 

 trifling importance. The first relates to a lottery which was started 

 in Paris in 1710, in which the projector had offered to the public 

 terms which were very disadvantageous to himself The second is 

 an easy exercise in combinations. The third relates to a game 

 called Le Jeu des Oublieux. The fourth is an extension of 

 Huygens's eleventh problem, and is also given in the Ars Conjee- 

 tandi, page 34. These four problems are new in the second edition. 



