a coin toss probability question I keep tossing a fair coin until I see HTHT appears. What is the probability that I have already observed the sequence TTH before I stop?
Edit:
Okay. This problem is the same as before. I am trying to think of the following. Could anyone please tell me if this is also the same as before, and if not, how to solve it.
Two players A and B toss two different fair coins. A tosses his coin until he sees HTHT appearing for the first time, and records the number of tosses, $X$. B tosses his coin until he sees TTH appearing for the first time, and records the number of tosses made, $Y$. What is the probability that $Y<X$?

By the standard techniques explained here in a similar context, one can show that the first time $T_A$ when the word A=HTHT is produced has generating function
$$
E(s^{T_A})=\frac{s^4}{a(s)},\qquad a(s)=(2-s)(8-4s-s^3)-2s^3,
$$
and that the first time $T_B$ when the word B=TTH is produced has generating function
$$
E(s^{T_B})=\frac{s^3}{b(s)},\qquad b(s)=(2-s)(4-2s-s^2),
$$
The next step would be to develop these as series of powers of $s$, getting
$$
E(s^{T_A})=\sum_{n\ge4}p_A(n)s^n,\quad E(s^{T_B})=\sum_{k\ge3}p_B(k)s^k,
$$
and finally, to compute the sum
$$
P(T_B<T_A)=\sum_{k\ge3}\sum_{n\ge k+1}p_B(k)p_A(n).
$$
An alternative solution is to consider the Markov chain on the space of the couples of prefixes of the words A and B and to solve directly the associated first hitting time problem for this Markov chain, as explained, once again, here.
(+1 to Arturo's and Mike's comments.)
Added later Introduce the decomposition into simple elements
$$
\frac1{(1-s)b(s)}=\sum_{i=1}^4\frac{c_i}{1-s\gamma_i}.
$$
Then, one can show that
$$
P(T_B<T_A)=\sum_{i=1}^4\frac{c_i}{a(\gamma_i)},
$$
by decomposing the rational fractions $1/b(s)$ and $1/a(s)$ into simple elements, by relying on the fact that all their poles are simple, and by using some elementary algebraic manipulations. 
At this point, I feel it is my duty to warn the OP that to ask question after question on math.SE or elsewhere along the lines of this post is somewhat pointless. Much more useful would be to learn once and for all the basics of the theory of finite Markov chains, for instance by delving into the quite congenial Markov chains by James Norris, and, to learn to manipulate power series, into the magnificent generatingfunctionology by Herbert Wilf.
