Let's say I have a normal, unbiased 6-sided die and I roll it 100 times. Some number is going to get rolled the same number or more often than any other number. What is the probability distribution and expected value for the number of times that that number gets rolled? How can we generalize this to a $k$-sided die that is rolled $n$ times?
For example, I might roll 14 $1$s, 10 $2$s, 20 $3$s, 16 $4$s, 23 $5$s, and 17 $6$s. In this case, the result I'm after is the number $23$, because that was how many times the most-rolled number ($5$) was rolled.
I know a few things about this distribution. It should be some sort of distribution over the range $[⌈n/k⌉,n]$ where it peaks sharply at the low end and has a long but vanishingly small tail. As $n$ increases, the density gets multiplicatively closer and closer to $⌈n/k⌉$. In fact, if we represent the expected value of this distribution as $c\cdot n/k$, then as $n$ approaches infinity, I believe $c$ should approach $1$. With this representation, I'm most interested in how to calculate $c$ for relatively small values of $n$ and $k$.
