User blog:DocTanner/Why ARAM doesn't feel random

Hi, DocTanner here with some math to lay on the line. If you're not mathematically inclined, feel free to move on after the tl;dr and just accept my judgement as True Fact. (Capital 'T'' and 'F' let you know it's Real.)

Too long, didn't read. (tl;dr)
Random champion selection does exactly what it says it does: randomly selects a champion.

The problem isn't the system, it's you. Yes you, the player. You're the one that's not random.

Why it's true
Okay, so over the past few weeks I've seen a lot of discussion and complaints about how the random champion selection in ARAM isn't actually Random. Specifically, I see these complaints most often:
 * Weighted towards free-list champions.
 * Weighted towards "ARAM-OP" champions, such as Nidalee.
 * Weighted towards "common" champions everyone owns, like Garen.
 * Weighted against "black sheep" champions that aren't played often.

Every one of these complaints is absolutely true.

"But, Doc... you just said it was random! How can all those things be true if it's a random system?!" you cry out from your cheeto-covered keyboard. It's okay fellow LoLlers, I'll explain. Remember how I said it was your fault? Well, here's why...

You don't own every champion. (Okay, maybe YOU do. But most people don't. ... in fact, most people remember what "outside" looks like. Remember? The place where the sun is?) That, right there, is the problem. If every player in an ARAM owned every champion, then the system would generate a completely random set of champions. But it can't, because you don't.

Simplified Example
First, let's simplify the problem. Let's say that there's 100 champions available. Then let's say that, on average, each champion is owned by 30% of players. This is, of course, a gross simplication, but it'll give us a good place to start.

With these numbers, each of the ten players owns 30 champions and each champion is owned by 3 players. Of the champions a player owns, each has a 1:30 (3.33%) chance of being selected. Among all three players owning this champion, this chance raises to ~9.67% chance of being selected.


 * Quick aside for those of you screaming that it's actually 10%, because three players makes it 3/30. Nope. Let's say we want Nidalee.


 * Player 1 has a 1/30 chance of getting her.
 * Assuming Player 2 doesn't own the champion Player 1 rolled, he now has a 1/30 chance. Adding their odds together, you'd get 2/30, right?
 * Yes, except that Player 2 only gets the chance if Player 1 didn't roll Nidalee. So Player 2's odds are actually (29/30)*(1/30) = 29/900.
 * Together, their odds are 59/900.
 * Finally, player three gets his chance and it works the same way: (841/900)*(1/30) = 841/27000.
 * Add his odds in with the total for 1770/27000 + 841/27000 = 2611/27000 = 9.67%
 * Of course, this increases slightly if someone rolls a champion owned by someone after them... but it's a small enough difference that it doesn't really matter.

Anyway, this makes sense, since there's 100 champions and 10 players. You'd think any given champion would have about 10% chance, right? Right.

... except now we get to mess up EVERYTHING. We're gonna add in the free list. Again assuming that everyone own 30% of the champions, they each already own 3/10 of the free list. So each one has an extra 7 champions in their pool. This means that any given non-free list champion is even less likely to get picked. It's the same math as before, except it's 1/37 instead of 1/30.

INTERMISSION I'm at work and need to change stations. So, I'll have to continue this later... ignore it for now.