What formula could help me quantify the probability of a chain of three letters (English Alphabet) where each letter is based on the previous one (stochastic modeling, Markov-chains, probabilities)
Example with the word "Bake" What is the probability of AKE, where A depends on B, K depends on A, and E depends on K.
Example with the word "State" What is the probability of ATE where A depends on T, T depends on A and E depends on T.
As the letters are not equiprobable as they depend on the previous one (as shown in the example).
Not much of a formula; really just an expression that encodes the rules you've described in your question:
Let $\alpha$ represent the start of the word, and $\omega$ represent the end of the word. Let $\Sigma = \{A, B, \ldots, Z, \alpha, \omega\}$ be the alphabet. Then we denote the letter-transition probabilities by $p_{\mu,\nu} \equiv P(\beta_{i+1} = \nu \mid \beta_i = \mu)$ for any $i$, where $\mu, \nu \in \Sigma$. Any given word $\beta_1\beta_2\cdots\beta_n$ then has the probability
$$ P = p_{\alpha,\beta_1} p_{\beta_n,\omega} \prod_{i=1}^{n-1} p_{\beta_i, \beta_{i+1}} $$
In general, there's no short cut. If the letters are independent, then we can replace $p_{\mu,\nu}$ by $p_{\nu}$, but otherwise, there's no additional short cut there, either.