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I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement.

The formula is applied B times (bootstraped iteration) with replacement in original populations between each iterations.

If I understood well, I can write that inside each iteration, probability for an element to be sampled P(e) is:

And probability of the whole vector S of size n to be sampled P(s) is:

But how to explain that between each iteration B, probabilities P(e) and P(S) are restaured to their origin ?

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Is it OK to say that and inside an iteration; and that between iteration? And what about P(S) ?

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