Expected number of draws of a labeled ball knowing it's already been drawn once

Question

Suppose you are drawing labeled balls from an urn with replacement, and you record what you've drawn. You have $n$ balls and draw $m$ times.
My question is, for the balls that you've drawn, what is the expected times you've drawn them? That is, if you point to one ball you recorded, what is the expected number of times you recorded it?

Answer 1

Outline: Let $D_i$ be the event Ball $i$ is drawn at least once. Let $W_i$ be the number of times Ball $i$ was drawn. We are looking for $E(W_i|D_i)$. Note that $$E(W_i|D_i)\Pr(D_i)=E(W_i).$$
To complete, use the fact that
$$\Pr(D_i)=1-\left(\frac{n-1}{n}\right)^m\quad\text{and}\quad E(W_i)=\frac{m}{n}.$$