Drawing n times from an urn that has m unique elements. Drawing one at a time, with replacement. What is the probability that resampling happens during a draw on average?

Clarification: The first draw has probability 0 of resampling. The second draw has probability 1/m of resampling. For further draws, it gets more complicated, because it matters whether and how often resampling happened during the previous draws.

I am interested in the average probability of resampling over n draws.