Proposition: Urn Model Without 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, without placing 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

\[p_k=p(A)=\frac{\binom Mk\binom {N-M}{n-k}}{\binom Nn}\]

Proofs: 1

  1. Problem: Broken Items in the Box II

Solutions: 1

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