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Originally Posted by SwordOrShield
Let's do some math. The chance of the leaver being on your team is 4/9. Let's stretch this out over 20 games, and see what the odds are that of, say, 10 leavers (Not too much of a stretch), the 'odds' of having them on your team 7 times out of 10 or more.
The odds of getting 7 is 7%. 8 is 2.1%. 9 and 10 together, about 0.4%. A total chance of about 9.5% here.
I see some problems with your math.
First of all, if you have a 4/9 chance that the leaver will be on your team, you have a 5/9 chance that the leaver will be on the other team. In other words, there's a chance that you'll win games that you would have otherwise lost and have your ELO artificially raised.
Second of all, leavers only matter when they're on your team several games in a row. Take a scenario when you have a leaver on your team, then no leaver, then leaver, etc. When you arbitrarily lose a game and your ELO is artificially decreased, then the next game you play will be against people below your level and you'll win, thereby raising your ELO back up.
Third an finally, I'm not sure where you got your percentages from. A streak of 7 leavers, using your 4/9 probability, has a chance of roughly .3%. That's only the probability that the leaver is on your team, if we take the chance of a leaver being in the game at all to be 50% then the probability of 7 straight leavers on your team drops off to near impossibility.
And even with all this, what does it really matter? So you lose a few straight games due to leavers then you just climb back up again. Your proof is meant to support the existence of ELO hell, but in fact it just relies on the assumption that it already exists. There is no consequence of the first 20 games unless ELO hell already exists to be trapped in.