We consider the case of a head and tail game. To do a proof around the Walter Penney paradox I have to use that $\mathbb{P}(T<\infty)$=1 where T is the random variable that modelizes the number of repetition before obtaining for the first time the sequence : head head tail.
The book I read states that to prove this point I have to use law of average ,but I don’t know what to do.
We know that $p:=P(HHT) \geq \frac{1}{2^3}$, so for a sequence of tosses of length $n$ the ratio of number of $HHT$ subsequences to non-$HHT$ subsequences will approach $p$ as $n\to \infty$.
Therefore, $HHT$ occurs infinitely often in an infinite sequence of coin tosses. For $P(T<\infty) < 1$ there must be a positive probability that we never see $HHT$, but we just showed that it occurs infinitely often and has positive probability.
Therefore, $$P(T=\infty)< \lim_{n\to \infty} \left(\frac{1}{2^3}\right)^n = 0$$
The inequality is due to the fact that I am only considering $HHT$ as independent events (i.e., infinite sequence of trials, where I flip a coin three times) vs the actual case where I can piggyback off the previous toss, and so only need $HT$ in the subsequent two tosses.