Proof

(related to Proposition: Urn Model Without Replacement)

According to the probability of Laplace experiments, we have to apply the formula

\[p(A)=\frac{\text{# possible outcomes of }A}{\text{# possible outcomes of }\Omega}=\frac {|A|}{|\Omega|}.\quad\quad( * )\]

Each outcome of the Laplace experiment described in the urn model consists of \(n\) balls drawn from \(N\) balls, while the order of balls drawn does not play any role. Therefore, the number of possible outcomes is given by the binomial coefficient. \[|\Omega|=\binom Nn.\]

Similarly, the number of possible ways of drawing \(k\) black balls out of \(M\) black balls is given by \(\binom Mk\). For each sample of \(k\) black balls chosen that way, there are \(\binom {N-M}{n-k}\) possible ways of drawing \(n-k\) white balls out of \(N-M\) white balls. Thus, according to the fundamental counting principle, the number of possible ways of drawing \(k\) black balls out of \(M\) black balls and, at the same time, drawing \(n-k\) white balls out of \(N-M\) white balls is given by

\[|A|=\binom Mk\binom {N-M}{n-k}.\]

It follows from \(( * )\) that

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


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References

Bibliography

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