Probability Question 297
Question: A bag contains p white and q black ball. Two players A and B alternately draw a ball from the bag, replacing the balls each time after the draw till one of them draws a white ball and wins the game. If A begins the game and the probability of A winning the game is three times chat of B, then the ratio p:q is:
Options:
A) 3 : 4
B) 4 : 3
C) 2 : 1
D) 1 : 2
Show Answer
Answer:
Correct Answer: C
Solution:
Probability of A winning [A can win in 1st or 3rd or 5th ?games if B loses 2nd or 4th or ?games] $ =\frac{p}{p+q}+{{( \frac{q}{p+q} )}^{2}}.\frac{p}{p+q}+{{( \frac{q}{p+q} )}^{4}}.\frac{p}{p+q}+… $
$ =\frac{\frac{p}{p+q}}{1-{{( \frac{q}{p+q} )}^{2}}}[ In\inf initeG.P.S=\frac{a}{1-r} ] $
$ =\frac{p(p+q)}{{{(p+q)}^{2}}-q^{2}} $ Probability of B winning $ =1-\frac{p(p+q)}{{{(p+q)}^{2}}-q^{2}}=\frac{{{(p+q)}^{2}}-q^{2}-p(p+q)}{{{(p+q)}^{2}}-q^{2}} $ Given $ P(A)=3P(B) $
$ \Rightarrow p(p+q)=3[{{(p+q)}^{2}}-q^{2}-p(p+q)] $
$ \Rightarrow 4p(p+q)=3(p+2q).p $
$ \Rightarrow 4p+4q=3p+6q\Rightarrow p=2p $
$ \frac{p}{q}=2 $ Or $ p:q=2:1 $
BETA
