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 Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition