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 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 &quot;the observed sequence $O = O_1, O_2...$&quot;
Thus as you say $Pr(sequence) = Pr(O) = Pr(O_1) * Pr(O_2)...$ according to my understanding of the quotes.