Well the last sentence of your post.
If you google "abiogenesis probability" you unfortunately get a load of stuff related to debates with Creationists, which is not what I'm looking for at all. But as far as I can tell (and please correct me if I'm wrong) there just doesn't exist any reliable estimate of what this probability is. We have a lower limit on what it can be (or upper limit, depending on how you're writing your probabilities) based on the probability of a random chance formation of the most basic form of "living" molecules, which seems so improbable as to have been unlikely to have happened anywhere ever, but other than that (again, as far as I can tell) we just don't know.
And we can't work out how likely life is to evolve based on the statistics, because we have no statistics, other than "it definitely happened at least once".
So yeah... to work out how often something has happened in the universe, you don't just need to know how big the universe is and how long it's been around, but also at least some basic estimate of the probability of the thing happening. Which we don't know at all. So this whole discussion is based around nothing more than "probability deduced by gut instinct".
As I've tried to point out, this is not me saying "we're definitely alone in the universe" because such a statement is equally as unfounded as "we're definitely not alone in the universe".