Statistics - A Mathematical Approach to Ranked and Luck

Lots of people believe that they always get the rotten luck.  They say that they always get the leavers, they always get the trolls, they always get the feeders.  Now, for sure, there is a small possibility that they are in fact getting massively bad luck.  Anything is possible and there is a chance, no matter how small, that one person queuing a lot of games can get bad luck despite the odds.  However, for the most part, this belief that one gets only trolls, AFKers and generally bad teammates is flawed when it is confronted with simple logic.

Let’s take a moment to look at the math.

To begin, we’re going to have to make the assumption that the LoL matchmaking system is NOT actively working against you.  This is a fairly safe assumption to make considering that it’s a program and has no ulterior motive.  In other words, there is no reason to believe that the system is purposefully working against anybody.

If you are part of a team of 5, let’s call this Team A, that means that there are 4 random people who you do not know and yourself (assuming you are not duo).  The other team has 5 random people who you do now know which we will call Team B.

Let us take the chance that a random person will AFK/Feed/Troll as X in terms of a percent.

On the other team, the chance that they will have no leavers/feeders/trolls is equal to 100%*(((100-X)/100)^5). For your team, assuming you yourself do not leave/feed/troll, the chance you will have no leavers/feeders/trolls is equal to 100%*(((100-X)/100)^4).

We know that (100-X)/100 is going to be equal to less than 1 because 100-X will be between 0 and 100 which divided by 100 will have to be between 0 and 1.  So, this value raised to the 4^th will actually be greater than this value raised to the 5^th.  

For example, let’s say that the chance of someone leaving is 10%.

100 – 10 = 90

90 / 100 = .9

.9^5 = .59049

.9^4 = .6561

When we multiply these values by 100% we get the chances for nobody on the other team to be a leaver/feeder/troll is around 59% while the chances of nobody on your team being a leaver/feeder/troll is around 65% or 66%.   What this means is that not only are you not more likely to get a negative member on your team, in fact you are less likely to get one because you know that you yourself will not be the negative member.

Conclusion:  If you yourself are a positive influence on the game, over a long period of games in which random chance is negated by general trends you will always be more likely to have a positive team than those that you fight against.

The only people that will see negative trends in their games more than other people at their ELO are people who are themselves negative as long as they play enough games to negate random chance and they are not some incredibly small statistical improbability.