probability of an entire sequence, not just one value

I am reading "Statistical methods in bioinformatics"

I came across this formula: As explained in the book O are the observations: My question is why the author only specified $$prob(O)$$ for a sequence of length=1 ?, I assume he is only talking about one observation, but generally we have a sequence $$O_1 ,O_2,O_3,...$$.

I assume that $$Prob(sequence)= P(O_1).P(O_2)...$$ but I am not sure.

• The capital non-indexed O seems to refer to the whole observed sequence (O = O1, O2, ...), so the probability P(O) is of observing the whole sequence of unspecified length. Or, am I missing something? Sep 2 '20 at 12:43

The author writes in the quote that $$O$$ is to be taken for the whole set of positions $$O_{i}$$.
In the extracts that you provide there is no reason to believe that it is only for a sequence of length 1. The author writes "the observed sequence $$O = O_1, O_2...$$"
Thus as you say $$Pr(sequence) = Pr(O) = Pr(O_1) * Pr(O_2)...$$ according to my understanding of the quotes.