I have two random strings over an $N$-letter alphabet: one is a shorter $M$-letter string, and one is a longer $L$-letter string. Assuming that two or more instances of the shorter string can overlap, what is the probability, in general, that the $M$-letter string appears $k$ times in the $L$-letter string?
To clarify the comment about overlapping short strings, consider two strings over a binary alphabet: a longer string "000000", and a shorter string "00000". Here we would say that there are two overlapping instances of the short string in the longer string, with a four-character overlap.