I'm interested in the probability of a die appearing to be biased when it is, in fact, fair. I'm trying to derive a result given, without proof, on YouTube: http://youtu.be/6guXMfg88Z8?t=1m29s
The basic idea is this: you suspect a fair 20 sided die to be biased. The video claims that, if you roll your dice 100 times, there is a 1-in-50 chance of you getting an excess of threes by pure chance.
I've tried to count the cases, but I'm getting into trouble. Let's say there are $k$ threes, then there are
$\frac{100!}{k! \ (100-k)!}$
ways of distributing the threes amongst the 100 throws. The next part is where I'm getting stuck. I need to count the number of ways of distributing the other 19 numbers amongst the remaining $100-k$ throws. I suspect that this might be related to the number of partitions of $100-k$.