probability of an entire sequence, not just one value
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.