If you browse all the way down to section 5.3 here (also included just the picture below), you'll see a plot of the probability that you'll get (n+1) consecutive heads before I get n consecutive heads (both of us tossing our own fair coins, so two independent sequences). As n becomes large, this probability seems to approach 1 in 3. Is there an intuitive reason for this?
If $n$ is large, the chance of a run of $n$ heads is very small, $2^{-n}$. We can view each flip as a try by me to start a run of $n+1$ heads, which happens with probability $2^{-(n+1)}$ and a try by you to start a run of $n$ heads, which happens with probability $2^{-n}$. Almost all flips will fail for both of us. We can ignore those. The chance that I win is then $\frac {2^{-(n+1)}}{2^{-n}+2^{-(n+1)}}= \frac{\frac 12}{1+ \frac12} = \frac 13$. The reason it starts lower is that we might both start a run at the same time, at which point you win because your run finishes first.