We have set of letters: $\{A,C,G,T\}$. We draw uniformly independently with returning $n \ge 7$ letters. Let $X_i$ be the result of $i-th$ draw. In other words: we have $n$ independent random variables with distribution $\mathbb{P}(X_i=A)=\mathbb{P}(X_i=C)=\mathbb{P}(X_i=G)=\mathbb{P}(X_i=T)=\frac{1}{4}$. Let $Y_n$ be random variable counting the number of pattern: $\left( A,?,A\right)$ in sequence $\left(X_1, \ldots, X_n\right)$ (where $?$ is one of the letters: $A,C,G,T$).
For example in sequence $ \left(A,C,A,T,A,T,G\right)$, $n=7$, we have $Y_7=2$.
Calculate $\mathbb{E}Y_n$.
I have no idea how to even start with this task. I need your help.
Thanks to @JMoravitz:
I define $Z_i$ such that $Z_i=1$ when pattern start at $i-th$ place and $Z_i=0$ otherwise. Then we have $$Y_n=\sum\limits_{i=1}^{n-2}Z_i$$ It's obvious that $\mathbb{P}\left(Z_i=1\right)=\frac{1}{4}\cdot 1 \cdot \frac{1}{4}$ and $\mathbb{E}Z_i=\frac{1}{16}$. $$\mathbb{E}Y_n = \mathbb{E}\sum\limits_{i=1}^{n-2}Z_i= \sum\limits_{i=1}^{n-2}\mathbb{E}Z_i = \frac{n-2}{16}$$