<s>[INST] Stuck on a probability problem/Expectation of coin toss I'm stuck on the following problem that is due for tomorrow:

We're flipping 1 coin indefinitely. $X$ is a random variable that count the amount of coin tosses.
  What is the expectation of the number of coin toss until we have the following sequence: THH (T: tail, H: head)

We know that TTT has an expectation of 14.
Edit: Someone told me that 
$E[X]= 3 + P[X>3]E[X].$
$P[X>3]=(1 - (1/8))$
$E[X]=24$
But I have now idea how to get to the last step?
 [/INST] Generating Function Approach
Duration Until $\boldsymbol{THH}$
Any trial can uniquely be constructed from any number of $H$ atoms, then an arbitrary combination of $TH$ and $T$ atoms, then a $THH$ atom. Put this together in a generating function:
$$
\begin{align}
\overbrace{\ \ \frac1{1-x}\ \ }^\text{$H$ atoms}\overbrace{\frac1{1-x-x^2}}^\text{$TH$, $T$ atoms}\overbrace{\quad\ x^3\quad\ \vphantom{\frac11}}^\text{$THH$ atom}
&=\frac{x^3}{1-2x+x^3}
\end{align}
$$
The expected duration is then
$$
\begin{align}
\left.x\frac{\mathrm{d}}{\mathrm{d}x}\frac{x^3}{1-2x+x^3}\right|_{x=\frac12}
&=\left.\frac{x^3(3-4x)}{\left(1-2x+x^3\right)^2}\right|_{x=\frac12}\\[9pt]
&=8
\end{align}
$$

Duration Until $\boldsymbol{THT}$
Any trial can be uniquely constructed from any number of $H$ atoms, then an arbitrary combination of $T$ and $TH^n$ atoms for $n\ge2$, then a $THT$ atom. Put this together in a generating function:
$$
\overbrace{\ \ \frac1{1-x}\ \ }^\text{$H$ atoms}\overbrace{\frac1{1-x-\frac{x^3}{1-x}}}^\text{$T$, $TH^n$ atoms}\overbrace{\quad\ x^3\quad\ \vphantom{\frac11}}^\text{$THT$ atom}=\frac{x^3}{1-2x+x^2-x^3}
$$
The expected duration is then
$$
\begin{align}
\left.x\frac{\mathrm{d}}{\mathrm{d}x}\frac{x^3}{1-2x+x^2-x^3}\right|_{x=\frac12}
&=\left.\frac{x^3\left(3-4x+x^2\right)}{\left(1-2x+x^2-x^3\right)^2}\right|_{x=\frac12}\\[9pt]
&=10
\end{align}
$$

Duration Until $\boldsymbol{TTT}$
Any trial can be uniquely constructed from any number of $H$, $TH$, and $TTH$ atoms, then an $TTT$ atom. Put this together in a generating function:
$$
\overbrace{\frac1{1-x-x^2-x^3}}^\text{$H$, $TH$, and $TTH$ atoms}\overbrace{\quad\ x^3\quad\ \vphantom{\frac11}}^\text{$TTT$ atom}=\frac{x^3}{1-x-x^2-x^3}
$$
The expected duration is then
$$
\begin{align}
\left.x\frac{\mathrm{d}}{\mathrm{d}x}\frac{x^3}{1-x-x^2-x^3}\right|_{x=\frac12}
&=\left.\frac{x^3\left(3-2x-x^2\right)}{\left(1-x-x^2-x^3\right)^2}\right|_{x=\frac12}\\[9pt]
&=14
\end{align}
$$
</s>