So much stupid.
{quoted}
That whole thing is a mess of stupid bullshit and gambler's fallacy. What we're talking about here are the odds of him having a troll on his team VS the odds of a troll being on the other team.
No, we are not talking about the odds that he has a troll on his team versus the enemy team, we are talking about the odds that he has more trolls than the enemy team. It's a subtle, but important difference. After all, why does it matter if OP has a troll on his team if the enemy also has a troll(s)?
This has no bearing whatsoever on anything. We are not calculating the odds of him having a troll on his team, we are calculating the odds of him having the odds on his team VS the odds of a troll being present on the enemy team. In a coin toss, the odds of a heads or a tails are equal, both 50%. Throw a one hundred sided die though and the odds of getting 1 or 100, though both 1%, are still equal. So, the odds of the presence of a troll in a game have no bearing whatsoever in the calculation.
Yes it does. If no one is a troll, then the game is pure skill. If everyone is a troll, then Op's team always has less trolls and should always win. The percentage of players that are trolls directly effects this. And again, we shouldn't care about one very specific situation (1 troll games). Instead we should care about the big picture, IE the chance that OPs team gets more trolls than the enemy team. In order to calculate that, we need to first determine the chance that OP gets multiple trolls and the chance that the enemy team gets trolls. From there we get the percent chance in an individual game that the enemy team gets less trolls than the OP's team.
Stop right there. This entire section is worthless math. Again, we are not calculating the odds of the presence of a troll, but the odds of the presence of a troll on his team VS the opposing team. With 4 possible trolls on his team VS 5 on the enemy team, it's 4/9 for a troll on his team and 5/9 for the opposing team. If you calculated the odds correctly, you would come to the same conclusion. But instead you shoved in the worthless "odds of a troll being present" statistic, which is worthless.
Again for the first time, you never had any statistics class. The relevant question is does OP get more trolls than the enemy team? And the answer is no. That is what the "worthless math" shows. But instead you want to focus on one troll games, which provides a meaningless snapshot. Not only is that snapshot meaningless, but you don't even do anything with that snapshot that resembles meaning. Essentially, you are saying that the coin will land on heads 4/9 (again, next to meaningless) of the time. You are looking at a single flip, while I am looking at 100 flips.
Not only is his result wrong because he used the wrong odds, but he doesn't know how binomial calculators work. First of all, if you enter his value, which is 0.29, the result you get is 0.000312%, not 0.000798%. The odds we want aren't >=50, but >50. Second of all, the way a binomial calculator works is by comparing the cumulative odds of success. As an example, the odds of a coin toss are always 50/50, but the odds of getting an equal number of heads and tails over a number of trials is not actually 50/50, but gets closer to the value as more trials are made.
I do know how a binomial calculator works. I made a claim that 50 or more games so I used the results relevant to 50 or more game IE >= 50.
Don't understand yet? All right. Go enter 0.5 on the calculator value, with 51% success rate. Start by typing 51 successes for 100 trials. Now, try 510 successes for 1000 trials. 5100 successes for 10000 trials. 51000 successes with 100000 trials. Notice how the odds keep dropping and dropping and dropping. This is because the odds of deviating from the norm go down as the number of trials go up. In turn, this does not affect the odds of the event happening, which is 50 fucking percent. What the calculator is calculating is not the odds of him getting a troll, but the odds that he'll deviate from these odds.
You don't even know how a binormial distribution works or what it does. If you have ever had a statistics class, I want you to go to your teacher, and tell them they failed you.
tldr: You don't understand math.
TLDR: You think you understand math, but you don't