Really? How many statewide math competitions have *you* achieved first place in? And I'll bet you $10,000 that your SAT math score is not higher than mine. Now that we've gotten that out of the way...
In the context of the discussion, Nebullis STIPULATED that we were talking about NONFUNCTIONAL mutations, as to which no natural selection could occur. So, there is no basis for assuming that the mutation proliferates. It could, by a random drift process, but just as likely the mutation would be snuffed out randomly.
Of course.
Hence, until natural selection occurs with respect to the mutation, the a prior probability that the mutation exists in Generation X is still the original probablity it occurs at all. So it doesn't matter when the mutations 1 to N occur. The probability is still the product of the individual probabilities PI.
Okay, I'll concede that since you didn't define your probability values, "P(i)", I had presumed that you were speaking of the "single instance" probability, and not the "long-term probability" of the mutation occurring. This is a natural mistake, since creationists have long been in the habit of making that error and I've lost count of the number of times I've caught them at it.
However, if that's what you actually meant, your original conclusion needs more development before you can actually state with any confidence that the combination of mutations would be, in your words, "next to impossible".
First, you throw around "N" as if it's some unspecified large number. Certainly, for a sufficiently large "N" a given combination of mutation becomes highly unlikely. But you fail to demonstrate that the number "N" in the case of several neutral mutations leading to a useful combination is necessarily a large one. "Lucky" combinations of 2-10 mutations are hardly astronomically unlikely, even if combinations of 20-100 of them are.
Similarly, I think you overstimate how unlikely a given mutation is if you're truly using the "long-term probability" value for your calculation. That probability includes the chances of the mutation happening in any individual of a lord-knows-how-large species population, multiplied by lord-knows-how-many generations over time.
You have yet to establish that N is sufficiently large, or P sufficiently small, to support your "next to impossible" assertion.
Finally, your P1*P2*P3...*PN calculation (as well as most other invocations of it in Creationist literature) makes the classic fallacy of discounting how many *other* combinations of mutations (at *any* location in the genome) might have been equally useful. It's easy to look at any particular sequence of mutations after the fact and say, "wow, the odds of *that* particular combination of mutations occurring is extremely low!" And so it is -- if you turned back the clock and "re-ran" the evolution experiment for that species, *that* exact sequence of (eventually) beneficial mutations would be highly unlikely to happen again in exactly the same way.
*However*, what this overlooks is the fact that there are nearly countless possible mutations which would have conferred some advantage on the species had they occurred. The odds of evolution hitting on *some* combination of mutations which gave an improvement (over a large chunk of time) are a damned sight better than P1*P2*P3...*PN. You can't argue that evolutionary improvement is mathematically unlikely by only looking at one *particular* improvement and showing the odds against it to be large, you have to look at the odds of getting *any* of the *large* number of mutations (or combinations) which would have resulted in increased fitness.
Let's say you're playing poker with three of your friends. Shuffle your deck of cards. Now write down the sequence of the cards in the deck. Congratulations, the odds against you getting that exact sequence are 80,658,175,170,943,878,571,660,636,856,404,000,000,000,000,000,000,000,000,000,000,000,000 to one. Did a miracle just occur for something so insanely unlikely to occur? No, since *some* combination was going to happen, and they're *all* that individually unlikely. If you deal those cards and win the pot, was that a 80,658,175[etc.etc.] to 1 miracle occurrence? No, since *many* different deck orderings could have let you win as well -- the actual likelihood of you winning a hand is actually far greater than the "P1*P2*P3" math of "how likely is that *one* deck arrangement" would lead you to believe -- in fact, it's one in four. But to use standard creationist math, you'd get beaten up by your friends for "cheating", because that *one* particular deck arrangement was *insanely* unlikely to occur...
You can't look at one particular evolutionary outcome and calculate the odds against that *exact* thing happening, because it artificially ignores the odds of *any* beneficial mutation happening. More concretely, one in a trillion odds for a particular beneficial mutation/combination aren't ridiculously low if there are five trillion possible beneficial mutations, out of 100 trillion possible mutations. It just looks that way if someone focusses on the "particular" odds and not the "overall" odds.