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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. In effect, I want to count the number of resamples in n draws and divide by n.

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  • $\begingroup$ Could you please explain the distinction you are implying between "probability" and "average probability"? $\endgroup$
    – whuber ♦
    Commented Feb 2 at 14:44
  • $\begingroup$ Which draw? This matters sine in the first draw there is 0 chance of resampling. Or are you asking what is the probability of getting "at least one resample during n draws"? $\endgroup$
    – user2974951
    Commented Feb 2 at 14:45
  • $\begingroup$ I understood the question as the second thing you wrote: at least one resample during n draws. But Dav should better specify what he needs. $\endgroup$
    – LevG
    Commented Feb 2 at 14:54
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    $\begingroup$ @LevG agreed. Please note your interpretation of this question already has answers in many posts here on CV, because it is easily answered by counting samples with and without replacement. $\endgroup$
    – whuber ♦
    Commented Feb 2 at 15:21
  • $\begingroup$ Thanks for your comments. I edited the question. @LevG I am not interested in "at least one resample during n draws", rather counting the number of resamples in n draws and dividing it by n. $\endgroup$
    – Dav Bhaji
    Commented Feb 2 at 16:32

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