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An urn contains $0$ black balls and $n$ white balls. $k$ balls are picked up at random and replaced by $k$ black balls. This process repeats itself till all the balls in the urn are black.

What is the expected number of black balls in the urn after $j$ steps of picking up?

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Focus on one ball. The chance that it has not been replaced after $j$ trials is $(1-{k\over n})^j$. The chance that it has been replaced is $1-(1-{k\over n})^j$, and so the expected number of replaced (black) balls after $j$ trials is $n\left(1-\left(1-{k\over n}\right)^j\right).$

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