A) \[\frac{1}{3}\]
B) \[\frac{1}{5}\]
C) \[\frac{1}{4}\]
D) \[\frac{2}{5}\]
Correct Answer: A
Solution :
X wins when the outcome is one of the following set of outcomes: H, TTH, TTTTH,??.. Since subsequent tosses are independent, the probability that X wins is \[p+\frac{p}{4}+\frac{p}{16}+....=\frac{4p}{3}\] Similarly \[Y\] wins if the outcome is one of the following: \[TH,TTTH,TTTTTH,....\] So, the probability that Y wins is\[\frac{1-p}{2}+\frac{1-p}{8}+\frac{1-p}{32}=\frac{2(1-p)}{3}\] Since X and Y win with equal probability, we have \[\frac{4p}{3}=\frac{2(1-p)}{3}\Rightarrow p=\frac{1}{3}\] So, option A is the correct answer.
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