LAPLACE. 533 



I. A player A draws a ball from an urn containing white 

 balls and black balls ; his chance of drawing a white ball is p, 

 and his chance of drawing a black ball is q : after the ball has 

 been drawn it is replaced. Then a second player B draws a ball 

 from a second urn contairing white balls and black balls; his 

 chance of drawing a white ball is p, and his chance of drawing 

 a black ball is q : after the ball has been drawn it is replaced. 

 The two players continue thus to draw alternately a ball, each 

 from his own urn, and to replace the ball after it has been 

 drawn. If a player draws a white ball he counts a point ; if he 

 draws a black ball he counts nothing. Suppose that A wants x 

 points, and B wants x points to complete an assigned set, required 

 the probabilities in favour of each player. 



II. Suppose A draws from an urn in which there are balls 

 of three kinds ; for a ball of the first kind he counts two points, for 

 a ball of the second kind he counts one point, and for a ball of the 

 third kind he counts no point: let his chances he p>,p)^, and q for 

 the three cases. 



Similarly let B draw from a second urn containing similar 

 balls ; let^', j9j', and q be his chances for the three cases. Then, 

 as before, we require the probabilities for each player of his 

 making up an assigned set of points before his adversary makes 

 up an assigned set. 



III. An urn contains a known number of black balls, and a 

 known number of white balls ; a ball is drawn and not replaced ; 

 then another ball, and so on : required the probability that a 

 given number of white balls will be drawn before another given 

 number of black balls. 



These three problems are solved by the method of Generating 

 Functions used carefully and accurately ; that is, the terms which 

 are required to make the equations true are given, and not 

 omitted. See Art. 97^. After the problems are solved generally 

 particular cases are deduced. 



The student of the fourth Supplement will have to bear in 

 mind that in the first problem p + q = l and p?' + 2=1* ^^^ ^^ 

 the second problem p +p)^ + ^ = 1, p' +p^ -\- q =1, 



