Given a dice with $m$ sides that is thrown $n$ times, what is the probability that $k <= m$ is the maximum number obtained?
Here is my attempt:
In order of $k$ to be the maximum, in at least one throw I need to get $k$ (and there are $n$ ways to get this) and in the rest of $n-1$ throws I can get anything between $1$ and $k$, and there are $k^{n-1}$ ways to do this. So, $nk^{n-1}$ times out of $m^n$ I get the maximum k. Of course, this is wrong, as I count duplicates as different throws.
Best Answer
You might find it easier to calculate
and then take the difference to give the probability that none of the throws are strictly more than $k$, and at least one of them is $k$.