Yes, we can calculate a probability.
No, not in any way that's better / different than the Drake equation.
You are (to put it gently) playing a little fast and loose with the term "first principles" ... There is a long way to go from any 'first principles' to planet formation. Actually, even galaxy formation is a long way away from giving useful information on planet formation. Read up on the timescales involved, and you might see some reasons why.
You correctly point out that only a couple decades ago the very existence of exosolar planets was debatable. It is not a binary condition: a few were found, and that was a big deal, but only a few. Then more were found, and they were accepted as a rarity. Now thousands have been found - to the extent that they are present in a significant fraction of the systems surveyed. Nobody flipped a switch such that we now know all there is to know about exosolar planets ...
Look into how many Earth-like planets we've found. None. It doesn't mean they're not there, just that our tools aren't up to it, yet. The better tools/techniques we use, the more we find. Planet hunting is still very much in its infancy (or 'early childhood', at least).
Each of these stages of observational discovery has an impact on theory. Planetary Dynamics, as a field, has had to respond to evidence of previously unimaginable systems - eg, to explain "Hot Jupiters".
So, I'm not sure how you think the result would be useful to guide (or why you tack on "binary stars and so on"), but unless you had a much more detailed question - no, this problem is not susceptible to any but the broadest probablistic analysis.
Oh, wait ... re-reading your question ... Yes. The probability of exosolar planets is 1. There are exosolar planets.
// // (Edit, responding to comment, but might get long-winded again ...)
That's pretty binary: either all stars have planets (so "yes, let's search") or none do (so, "no, don't bother"). And if it's something in between? Let's say 1/3 stars have planets ... OK, what does that tell you? "Look over ... there ... and we'll see planets - maybe."
I don't mean to be curt, but these are very broad questions, and you seem to be assuming (optimistically) a lot about the state of the field. What I was trying to say above is that a) what you're asking for cannot be calculated in a useful way and b) you would need to ask a much more detailed question to get a useful answer. (I get the sense that English is not your first language, so I'm sorry if some of this is just a language issue.)
For example, a more useful question might be "Where are we more likely to find planets with higher levels of heavy elements?" This would be more useful in the sense that it would tell us what kinds of areas to search for planets that could support recognizable life. It's still a very difficult question to answer. Also, it's not as useful as you might think/hope - missions like Kepler have no shortage of interesting targets; we would need a lot more telescopes (and a lot more observations already completed) before we had to start narrowing down our choices like that.
Anyway, that would be (will be) answered observationally - by finding a lot of planets, and saying "We've surveyed this many sample areas, and seen this many of these kinds of planets in this kind of area, so in the future we expect that we'll have the same probabilities."
For example, "We've surveyed 100,000 stars at random in 200 areas of the Milky Way divided into 20 categories (such as 'mid-radius, middle of a spiral arm'), and found 3,000 planets. We were able to roughly categorize about half the planets. From our data, we hypothesize that in a galaxy of similar age and morphology to the MW, 2% of mid-mass stars in an outer spiral arm are likely to have planets." Then you do even more observation to see if that seems right. Note that to get a good model, you need to search everywhere; you need to check areas where you expect a 'negative result' to make sure that there are really no planets there.
Computations (simulations) sometimes give us interesting ideas, but usually they test our current understanding. We use simulations to see how well our theories fit the data we see; our theories are not absolute laws from which we know what we 'should' see. New data comes in, theories are either confirmed or modified, simulations are written to test what we think we understand.
If someone had tried to calculate 'the probability of extrasolar planets' 30 years ago, he would have done one of three things: a) obtained a result that agreed with data at the time (and been wrong), b) obtained a result that agreed with current data (and probably not believed it, had no way to confirm it - and possibly still been wrong), or c) obtained a result that agrees with the data we'll have in another 30 years (see previous).