What gives validity to the fine-tuning argument, I think, and separates what it is describing from other events in the universe is the definition of design, which entails things being Specifically Arranged for a Purpose. This is what allows us to distinguish between a crime scene and an accident and, yes, probabilities enter in.
When someone is found dead, the police show up and start looking around to determine first if it was an accident or if there was “foul play.” How do they know foul play when they see it? They look for things having been “arranged for a purpose” (in this case the purpose being to bring about someone’s death). So, if they find that the person was killed by rat poison and they find that the spouse had bookmarked several websites about poisons and then they find a receipt showing that she had bought rat poison and they find rat poison in the cheerios, which happens to be the victim’s favorite cereal, and so on, then they become suspicious. And yes, they are counting. As each “part” in this arrangement lines up toward the End (purpose) of getting this person dead, the detective concludes “murder” and makes an arrest. Could those things have happened by some kind of random chance? Of course, but any reasonable person would respond with a loud, “Unlikely!” (which is a probabilistic term).
The problem with the philosopher Ric Machuga’s “1 out of 1” rebuttal (the probability of something that's already occurred is 1) is that if applied consistently, detectives would be out of line making arrests and juries could not convict. The defense could simply assert that sure this LOOKS like a murder, but this was the 1 time when everything lined up just right by random chance but really there was no mind behind it.
Here’s another, and I think, better example: Say Joe goes to play the lottery and wins! Despite the odds being really low that Joe should win, no one claims a miracle or thinks that there is anything but randomness behind Joe winning. But then, Joe wins again… eyebrows would raise. If Joe won a third time, there would be an investigation! Why? These are all highly improbable events. Why the big change in our thinking as Joe goes from winning once to winning 3 times? The reason we aren’t suspicious that Joe won once is because we know that SOMEBODY HAD TO WIN. It’s a Lottery. People enter and someone always wins. Even though the chances of JOE winning are low, the chances of SOMEONE winning are 1 out 1. It just happened to be Joe. So while we think Joe is very fortunate, we don’t suspect a mind behind his winning. But when he wins twice and then three times, we get suspicious because while we know that it’s a 1 out of 1 chance that SOME person will win, we know the winner should be random… but this is not looking random… this is looking designed… that is, ARRANGED for a PARTICULAR PERSON (Joe) to win. The best explanation for Joe winning once is just dumb luck. The best explanation for Joe winning 3 times is Design, a specific arrangement of things for a particular purpose or function.
The fine-tuning argument notices that things seem specifically arranged for a purpose: To produce and sustain life. It looks more like a crime scene than an accident. Yes, the arrangement of Mt Lassen’s molecules at this very moment is a once in a lifetime, improbable event and yet it happened. But it is not arranged THAT way for THIS purpose and so it does not impress (except maybe aesthetically, but that’s another story – and it can have many many many different arrangements and still impress that way; but life’s parameters are narrow, as are the parameters of all designed things). In contrast, the arrangement of the universe, its constants and properties, initial conditions, the earth, the solar sytem, etc , etc are such that it seemed to have had “life in mind” which leads one to reasonably conclude that indeed a mind was at work. Mt Lassen won the lottery once. How nice. It looks like Joe (Life) won the lottery many many many times over.