168 DE MOIVRE. 



803. The general problem relating to the Duration of Play 

 may be thus enunciated : suppose A to have m counters, and B 

 to have n counters ; let their chances of winning in a single game 

 be as a to Z> ; the loser in each game is to give a counter to his 

 adversary : required the probability tliat when or before a certain 

 number of games has been played, one of the players will have won 

 all the counters of his adversary. It will be seen that the words 

 in italics constitute the advance whi'ch this problem makes beyond 

 the more simple one discussed in Art. 107. 



De Moivre's Problems LVIII. and Lix. amount to solving the 

 problem of the Duration of Play for the case in which m and n 

 are equal. 



After discussing some cases in which n = 2 or 3, De Moivi*e 

 lays down a General Rule, thus : 



A General Rule for determining what Probability there is that 

 the Play shall not be determined in a given number of Games. 



Let 71 be the number of Pieces of each Gamester. Let also n-hd 

 be the number of Games given; raise a + h to the Power n, then cut off 

 the two extream Terms, and multiply the remainder by aa + 2ab + hb : 

 then cut off again the two Extreams, and multiply again the remaiiM^er 

 by aa + 2ab + hb, still rejecting the two Extreams; and so on, makiog 



as many Multiplications as there are Units in ^d ; make the last Pro- 



duct the Numerator of a Fraction whose Denominator let be (a + b)"'^'\ 

 and that Fraction will express tlie Probalnlity required, ; still ob- 

 serving that if d be an odd number, you wj-ite d—1 in its room. 



For an example, De Moivre supposes n = 4<, d= 6. 



Raise a+h to the fourth power, and reject the extremes ; thus 

 we have 4<a^b + MV + ^a¥. 



Multiply by a^ + 2ab + V^, and reject the extremes ; thus we 

 have l^a'h' + 2^a%' + Ua^h\ 



Multiply by <^ + 2ah + W, and reject the extremes ; thus we 

 have 48a'Z>' + Q%a%' + ^Mh\ 



Multiply by a^+2«& + Z>^ and reject the extremes; thus we 

 have lUa%' + 232a'Z>'^ + l(j^a'h\ 



Thus the probability that the Play will not be ended in 

 10 games is 



