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

Question

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

Infinity Learn · Accepted Answer

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:$$b=2$$:$$1$$

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