."> 4 BOOK OF MATHEMATICAL PROBLEMS. 



247. A bag contains ten balls each equally likely to be white 

 or black : three balls being drawn turn out two white and one 

 black ; these are replaced and five are then drawn, two white and 

 three black : prove that the chance of a draw from the remaining 



71 

 five giving a white ball is y^ 



248. From a very large number of balls, each of which is 

 equally likely to be white or black, a ball is drawn and replaced p 

 times and each drawing gives a white ball : prove that the chance 



of drawing a white ball at the next draw is - _ 



249. A bag contains four white and four black balls ; from 

 these four are drawn at random and placed in another bag ; three 

 draws are made from the latter the ball being replaced after each 

 draw, and each gives a white ball : prove that the chance of the 

 next draw giving a black ball is "33. 



250. A bag contains m white balls and n black balls, and from 

 it balls are drawn one by one until a white ball is drawn. A bets 

 B at each draw x : y that a black ball is drawn : prove that the 

 value of A'a expectation at the beginning of the drawing is 



"-*--*. 



m + i 



251. From an unknown number of balls, each equally likely 

 to be white or black, a ball is drawn and turns out to be white; 

 this is not replaced and 2n more draws are made, the balls 

 being not replaced. Prove that the probability that in the 



2n + 1 draws more white balls are drawn than black is 



