OP ARTS AND SCIENCES. 141 



VI. 



PROBABILITIES AT THE THREE-BALL GAME OF 

 BILLIARDS. 



By Benjamin Peirce. 



Read Oct. 10, 1877. 



In the three-ball game of billiards, the person who makes a success- 

 ful shot adds one to his counts. In case of a discount, the person who 

 gives the discount loses, moreover, one from his count, when his oppo- 

 nent makes a successful shot. In the case when he gives a double 

 discount, he loses two for each successful shot ; and, in the same way, 

 for a treble, quadruple, &c., discount, he loses three, four, &c., points 

 from his count. In the grand discount, he loses all which he may have 

 madcj whenever his opponent succeeds in his shot. Whenever a 

 player fails in his shot, the other player takes the cue. 



Let the two players be A and B, and let h be the whole number of 

 points of the game. Let a be the probability that A will make- his 

 shot, and b the probability that £ will make his shot. No allowance 

 is made for the increase of probability of a successful shot after the 

 first shot, although this is a very important consideration with good 

 players. It may justly be thought that the failure to recognize this 

 change of probability reduces the practical value of the investigation. 

 But imperfection is inevitable in the earlier stages of any research. 



Let, then, A = — — -, B = —— -, 



a -\- b — ab a -\- b — ab 



80 that Ab= Ba = A-{- B — 1, 



Let A give n discounts to B. When A needs i more points to make 

 the game, and B needs _/ more points, let 



F{i,j) be A's probability of winning when he has the play, and 

 f{hj) be A's probability of winning when B has the play. 



