@hiyouthere:
Cool analysis. By adding more copies, you're scaling the probabilities that these different draws happen. Your analysis does a great job at doing that, but it doesn't address the main debating point: Is it worth giving up some probability of drawing a Vial in order to avoid drawing multiple Vials.
You can show another spreadsheet to compile this Bayesian problem (I had done it earlier), but the reader's digest version is this:
If 2 Vials in hand is better than 0 Vials, then you definitely want 4 Vials maindeck, since multiples don't hurt that much. Merfolk is an example of this. They run a lot of dudes, can really abuse the tempo off of Vial, and with cards like Standstill can abuse it further, so they would probably rather even see 2 Vials than 0 in a game.
If 2 Vials in hand is equal to or slightly worse than 0 Vials, then you probably want 3-4 Vials maindeck. In this case, you're just trying to maximize the probability of drawing exactly 1 Vial in a reasonable period of time, and the function is maximized at just over 3 Vials. But since you obviously can't pick a fraction of a Vial, we'll go with 3 or 4.
If 2 Vials in hand is quite a bit worse than 0 Vials, then you probably want 2-3 Vials maindeck, since this puts more weight on avoiding multiples.
The more multiple Vials hurt relative to not having any, the fewer copies you want. The better Vial is (i.e. the difference between exactly 1 and 0), the more copies you want, but the key difference is really the difference between multiple copies of Vial and zero copies of Vial.
An extremely good card that's so bad in multiples that you'd rather not see any than see multiples of it should probably go in between 2 and 3 copies. Vial falls into that category.
Another example that uses the same logic (though I haven't seen someone really think it through to this level) is Fireblast in Sligh. You really, really, *really* don't want to see two copies, but seeing one copy is amazing. What do you do? You run 2 copies.
Or Sylvan Library in Zoo is the same thing. One copy is absolutely amazing, in any matchup. But the second copy is completely dead. Drawing 2 is worse than not seeing any, so 4 copies is wrong and 2-3 copies is right. Further studies show that 3 copies is probably the right call.
Of course, in the case of Vial this is greatly complicated by the fact that Vial is very sensitive to the number of creatures that you're running in the deck. UW Tempo runs 15-17 effective vialables (Fathom Seer isn't very vialable). If you cut a Vial, it's replaced by some creature that can be vialed in, making the remaining Vials stronger.
4 Vials is almost certainly wrong because you would dip down to 15 Vialable creatures, which really defeats the purpose of wanting Aether Vials so much. With just 15 Vialables, Vial would be noticeably weaker.
Likewise 1 Vial is also almost certainly wrong because Vial is a very strong card and you can definitely get away with 2 copies without having to see much (if any) multiple draws. Playing only 1 doesn't take advantage of Vial's power.
But between 2 and 3 Vials, it's a pretty tough call. Jeff prefers a strong endgame, more options, and fewer risks. I'm more of the gambling type (I have to be based on my play skill -- if I don't take risks, I'd never win a game), so I'll take the slightly more risky gamble of running three copies so that I'll draw both more single copies of Vial and more multiples. It definitely makes the deck easier to play for me, particularly since I'm a more aggressive early-midgame player, I'm happy with three Vials.
