What is the probability that $THTH$ occurs before $HTHH$ in an infinite sequence of coin flips? 
What is the probability that $THTH$ occurs before $HTHH$ in an infinite sequence of coin flips? 

The expected number of flips until you first see $THTH$ is $6$, while the expected number until you first see $HTHH$ is $10$. Intuitively, I would guess that the probability that $THTH$ occurs before $HTHH$ is $3/4$. Is there a formal argument to compute this probability?
Probabilistic model: We denote by $(X_{n})_{n \geq 1}$ the random variable of coin flips, taking values in $\{H,T\}^{\mathbb{N}}$. We let $t_{THTH}$ be the first time that $THTH$ occurs in the sequence. We have
$$
\mathbb{E}[t_{H}] = \frac{1}{2} \mathbb{E}[t_{H} | X_{1} = H] + 
\frac{1}{2} \mathbb{E}[t_{H} | X_{1} = T] = \frac{1}{2}  + 
\frac{1}{2} (\mathbb{E}[t_{H}] + 1  ).
$$
$$
\mathbb{E}[t_{TH}] = \frac{1}{2} \mathbb{E}[t_{TH} | X_{1} = H] + 
\frac{1}{2} \mathbb{E}[t_{TH} | X_{1} = T]
$$

We have states $\{H,T,HT,TH,THT,HTH,THTH,HTHH\}$ which are the prefixes of your strings. We can write down a Markov transition matrix for these states:
$$A = \frac{1}{2} \begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{pmatrix}$$
This is an absorbing Markov chain already in canonical form \begin{pmatrix}Q&R\\0&I\end{pmatrix} thus we can find its fundamental matrix $N$ with
$$N = (I - Q)^{-1}$$
$$N = \frac{1}{7} \begin{pmatrix}
24 & 20 & 12 & 10 &  8 & 6\\
16 & 32 &  8 & 16 & 10 & 4\\
10 & 20 & 12 & 10 &  8 & 6\\
16 & 18 &  8 & 16 & 10 & 4\\
 8 & 16 &  4 &  8 & 12 & 2\\
 4 &  8 &  2 &  4 &  6 & 8
\end{pmatrix}.$$
Finally we can get the probabilities of our absorbing states given an initial probability vector of non-absorbing states $\vec p$with ${\vec p}^TNR$. Assuming we threw one coin already our initial state is $[1/2, 1/2, 0, 0, 0, 0]^T$, and thus our final answer is that we end up with $THTH$ $9/14$th of the time and $HTHH$ $5/14$th of the time.
