Expected number of times random substring occurs inside of larger random string

Question

I have a four-letter alphabet containing A, B, C, and D. What is the expected number of times a string of length $m$ occurs inside of larger random substring of length $n$, both generated from the same alphabet? I think I've got it so far for an even distribution, where each letter has a probability of $0.25$:

$$(n-m)\cdot\left(\frac 1 4\right)^m$$

What if the letters are not evenly distributed? What if A and B had probabilities of $0.115$, and C and D had probabilities of $0.385$? How does that change the problem?

Answer 1

You miscounted by $1$: A string of length $m$ fits into a string of length $n$ once, not $0$ times, so it should be $(n-m+1)$.

The problem is essentially the same if the probabilities differ. By linearity of expectation, you still only have to take $n-m+1$ times the probability for a single match. If you have $k$ letters $a_1$ through $a_k$ that occur with probabilities $p_1$ through $p_k$, the expected number of matches is

$$ (n-m+1)\left(\sum_{i=1}^kp_i^2\right)^m\;. $$