Mathematics - Probability Question with Solution | TestHub

If the outcome is one of the following: $H,TTH,TTTTH,\dots$, then $X$ wins As subsequent tosses are independent, so the probability that $X$ wins is $$p+\frac{p}{4}+\frac{p}{16}+\dots =\frac{4p}{3}.$$ Similarly $Y$ wins if the outcome is one of the following: $TH,TTTH,TTTTTH,\dots$ Therefore, the probability that $Y$ wins is $$\frac{1-p}{2}+\frac{1-p}{8}+\frac{1-p}{32}=\frac{2(1-p)}{3}$$ Since, the probability of winning the game by both the players is equal then, we have $$\frac{4p}{3}=\frac{2(1-p)}{3}\Rightarrow p=\frac{1}{3}$$