Suppose we have $N$ kinds of colored balls, and there are an infinite number of each kind. Now we'd like to draw $m(m>N)$ balls from them. How to calculate the probability P(m,n), which is the probability of drawing $n$ kinds of ball$(n\le N )$?
Below is my solution, which is only a recursive expression, I'm not sure whether it is true, and I hope someone can give me a closed form expression.
=my solution=
For $m
For $n=1$, $P(m,n)=(\frac{1}{N})^{m-1}$
For $n>1$, $P(m,n)=P(m-1,n)*\frac{n}{N}+P(m-1,n-1)*\frac{N-n+1}{N}$
The first and second expression are quite obvious. for $P(m,n),m\ge n,n>1$, consider the first $m-1$ balls we drawn, if we've drawn $n$ kinds of balls, then the last ball we draw must belongs to the $n$ kinds of balls. Otherwise, the last ball must be chosen from the left $N-n+1$ kinds.
Thank you!