Proposition: Urn Model With Replacement

An urn contains \(N\) colored balls, thereof \(M\) black balls with \(1\le M\le N\), and \(N-M\) white balls. All balls are well-mixed together. We draw randomly \(n\) balls (\(n\le N\)) from the urn, and place them back in the urn. We observe the colors of all balls drawn.

Assuming that drawing each ball is an elementary Laplace experiment, the probability \(p_k\) of the event

\[A:=\{\text{"If we draw }n\text{ balls in total, then we draw exactly }k\text{ black balls."}\}\]

obeys the equation

\[p_k=p(A)=\binom nk \left(\frac MN\right)^k\left(1-\frac MN\right)^{n-k}.\]

Proofs: 1

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  1. Bosch, Karl: "Elementare Einf├╝hrung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition