P(H) = p
P(T) = (1-p)
Winning sequence possible,
HH , THH, HTHH, THTHH, HTHTHH......
So respective Probabilities Sequence,
$p^2$, $(1-p)p^2$, $p(1-p)p^2$, $p(1-p)^2p^2$, $p^2(1-p)^2p^2$.....
So Probability to win =
$p^2$ + $(1-p)p^2$ + $p(1-p)p^2$ + $p(1-p)^2p^2$ + $p^2(1-p)^2p^2$ +.....
= $p^2$[1 + $(1-p)$ + $p(1-p)$ + $p(1-p)^2$ + $p^2(1-p)^2$ +......]
= $p^2$[(1 + $(1-p)$) + $p(1-p)(1 + (1-p))$ + $p^2(1-p)^2(1 + (1-p))$ +......]
= $p^2$[$(2-p)$ + $p(1-p)(2-p)$ + $p^2(1-p)^2(2-p)$ +.......]
= $p^2(2-p)$[1 + $p(1-p)$ + $p^2(1-p)^2$ +.......]
= $p^2(2-p)$×$\frac{1}{1 - p(1-p)}$
[Since sum of infinite G.P is $\frac{a}{1 - r}$ , here $a$ is 1 and r is $p(1-p)$ ]
= $\frac{p^2(2-p)}{(1 - p + p^2)}$