Introduction[]
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.)
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 that "black sheep" champion you really want to play.
How can a system that's supposed to be random cause all of these problems?
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[]
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, so maybe YOU do. But most people don't. ... in fact, most people remember what "outside" looks like. Remember? The place where the sun is?
Anyway, 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.
Math Example (Point 1)[]
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 simplification, 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
some tiny evil.
- Player 1 has a 1/30 chance of getting him.
- 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?
- Well, no, because Player 2 only gets the chance if Player 1 didn't roll him already. So Player 2's odds are actually (29/30)*(1/30) = 29/900.
- Between them, the odds of getting some evil is therefore 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 show up in the game: 7.89%, to be exact.
Okay, so more champs makes any given one less likely, that makes sense. But what about the free list? How likely is any given free list champion to show up in a game? 23.97%, that's how likely. That's right, if every one in the game owns 30/100 champions, and only 3/10 of the free list... then each free list champion has a nearly 1/4 chance of being in the game.
Conclusion: The free list shows up more often because they are available to people who otherwise wouldn't be able to play them.
Table Example (Points 2-4)[]
This time you don't need to try and figure out math. All you have to do is look at a table. Don't worry, you'll do fine. If you get confused, tt's all just the same math as before except this time I let Excel do all the work for me.
Here's how the table works: the percentages are the odds of a given champion appearing in the game if Y number of players own that champions and have a total of X champions available to them.
| Players | 30 Champs | 37 Champs | 50 Champs | 75 Champs | 115 Champs |
|---|---|---|---|---|---|
| 1 | 3.33% | 2.70% | 2.00% | 1.33% | 0.87% |
| 2 | 6.56% | 5.33% | 3.96% | 2.65% | 1.73% |
| 3 | 9.67% | 7.89% | 5.88% | 3.95% | 2.59% |
| 4 | 12.68% | 10.38% | 7.76% | 5.23% | 3.43% |
| 5 | 15.59% | 12.80% | 9.61% | 6.49% | 4.27% |
| 6 | 18.41% | 15.16% | 11.42% | 7.74% | 5.11% |
| 7 | 21.13% | 17.45% | 13.19% | 8.97% | 5.93% |
| 8 | 23.75% | 19.68% | 14.92% | 10.18% | 6.75% |
| 9 | 26.30% | 21.85% | 16.63% | 11.38% | 7.56% |
| 10 | 28.75% | 23.97% | 18.29% | 12.56% | 8.36% |
For example: If there are six players in the game, all of whom own 
Sona and 49 other champions, there is an 11.42% chance that she'll be in the game. Got it? Good.
So what does this mean? Well, first, let's just emphasize that last point about the free list. If a champion is available to all ten players, the odds of that champion appearing in game goes up dramatically if those players own fewer champions.
What does this say about the other three complaints, though? Let's address them one at a time.
Point 2: ARAM-OP[]
Why do some champions, such as kitty heal-spears, show up so often in an ARAM game? It's because of that guy. No... over there, the other one.
That guy, of course, is the guy with an ARAM-only account. It's an account that he created purely to play ARAM and it only has his favorite champions for the map. If he only has 20 champions, then he's got a 3.33% chance of rolling any given one even if there's no overlap with the free list. (20 + free list = 30, in case you're confused.) But you want Miss Kitty too! ... shame you have every champion, because your odds of getting her is 0.87%
So the whole thing is due to something that should be obvious: the fewer champions you have access to, the more likely you are to get the one you want. And that guy with 20 champions hand-picked for ARAM? Yeah, he's got a 2/3 chance of getting one of them. Throw a few of that guy in a game and, voila, team-comp OP.
Point 3: Common Champions[]
Everyone that isn't me owns Garen. Everyone. Even you. Seriously, go look, I bet you own him now. It's okay, I'll wait.
Back? Good, let's continue. Certain champions are just more common than others. That means that more players in any given game have access to them. Pick a column. The odds go up as more players get a crack at it.
Need a solid example? Okay. This time everyone's dropped some serious IP and is rocking 75 champions each. But poor little 
used-to-suck is only owned by three of them. 
League of Draven is a house favorite owned by all ten. 3.95% says you see the under-appreciated mage, while the axe-man sits pretty at 12.56% and over three times more likely to show up.
Point 4: Black Sheep[]
Go back and read point 3 again. Now pretend you like Karma. Good, you just figured out point 4.
Conclusion[]
ARAM is random, but that doesn't mean every champion is just as likely to show up as any other. It depends on how many players have that champion, and how many others they have available.
Every pattern and bias that you see in the ARAM system is either a statistical effect that is your fault or a figment of your imagination. It's kinda like when you complain about the ELO system "always matches you with better opponents than allies". ... don't even get me started about that one.
That's all for now. Please feel free to leave any comments you like. It's okay if you're not good at math, but be polite to each other.
Good luck, and have fun! --DocTanner (talk) 13:27, October 2, 2013 (UTC)
p.s. Thanks go to User:Luckyvampire who was kind enough to fix some spelling issues.