# Some fun maths.

Let's say you consistently and without fail play two ranks above your current rank. Thus, you should climb. Let's also say that the chance any player in your current rank is a troll, afks, feeds their lane, whatever is 10% or 0.1.
It follows that the chance you have one of these players on your team is the following:
>0.1\*0.1\*0.1\*0.1 + 0.1\*0.1\*0.1\*0.9 + 0.1\*0.1\*0.9\*0.9 + 0.1\*0.9\*0.9\*0.9 OVER 0.9\*0.9\*0.9\*0.9
Or about 12.498%. This is because if you play consistently, then there is no chance you are a troll, thus we only have to consider your four teammates. We add up all possible outcomes (the chance 4 teammates are trolls, the chance 3 are, etc.) And divide by the chance that *none* is a troll to get the ratio of games you will have at least one troll in.
Now, let's consider the chance those players are on the enemy team:
>0.1\*0.1\*0.1\*0.1\*0.1 + 0.9\*0.1\*0.1\*0.1\*0.1 + 0.9\*0.9\*0.1\*0.1\*0.1 + 0.9\*0.9\*0.9\*0.1\*0.1 + 0.9\*0.9\*0.9\*0.9\*0.1 OVER 0.9\*0.9\*0.9\*0.9\*0.9
This amounts to 12.499%. **This statistical fact holds true no matter what the proportion of trolls is in your rank.** The likelihood of a troll on the enemy team will **always** be higher than on your team, which means over hundreds of games, you **will** climb.
But we started off assuming that you *weren't* climbing - we assumed that you play above your rank and the only reason you aren't climbing is because of these trolls. We have proven that trolls have no factor on your rank, and that in fact you will win 0.001% more of your games **simply because the enemy has a troll** than you lost because you have a troll.
As such, there is a contradiction - we've proven that trolls actually make you win more games than they make you lose, which makes you climb, yet we assumed that you were not at your deserved rank and not climbing *because* of trolls. This cannot be the case, and thus one of three things must be true:
* You are not *not* at your deserved rank, thus you *are* at your deserved rank.
Or
* You are not climbing for some reason other than trolls (or afks whatever). Well the only thing other than your teammates that affects your climb is you... thus we must assume you are not climbing because of you...
Or
* You yourself are a troll, and thus our assumption you are not in our maths is wrong, thus the maths failed. Though, if you're a troll, there's a 100% chance of a troll on your team, and thus that's the reason you lose so..
Quod Erat Demonstratum. Trolls are not a factor in your rank.
****
#Limitations:
* Doesn't factor into account that this is statistics of infinity, thus obviously some players will experience differently, especially over 10s of games. However, the more games someone plays, the more closely this will approximate their teams, thus the less chance that trolls are actually impacting them negatively is.
* Doesn't take into account that the player in our case, the one we said was consistently two ranks above their skill, is not consistent, period. No player plays exactly the same each game for a variety of factors. However, this is a again a reason that can only be blamed on the player, not others.
* Does not take into consideration the chance another player in the game is playing above or below their skill. The math to prove this is exactly the same, but requires considering upwards of 2^9 configurations if we assume the player is constant or 2^10 configurations if we factor in the chance the player plays above or below skill. That's way too long for a post and honestly I can't be arsed to type it out, so I'll leave that to you to prove for yourself if you so desire.
There are some neat tricks to control the number of cases you have to deal with in this analysis and make it manageable, and if you look closely, you'll see that the proof is exactly the same thus the result is the same.
Good luck on the rift!

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