A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag, replacing  the ball each time after the draw till one of them draws a white ball and wins the game. A begins the game. If the probability of A winning the game is three times that of B, the ratio a:b is

Let W denote the event of drawing a white ball at any draw and B that for a black ball. ThenP(W)=aa+b and P(B)=ba+bP(A wins the game)=P(W or BBW or BBBBW or …)=P(W)+P(BBW)+P(BBBBW)+⋯=P(W)+P(B)P(B)P(W)+P(B)P(B)P(B)P(B)P(W)+⋯=P(W)+P(W)P(B)2+P(W)P(B)4+⋯=P(W)1−P(B)2=a(a+b)a2+2ab=a+ba+2b Also P(B wins the game )=1−a+ba+2b=aa+2bAccording to the given condition,a+ba+2b=3ba+2b⇒a=2b⇒a:b=2:1