# Proof

(related to Proposition: Urn Model With 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 drawing a ball $$n$$ times, while each time there are $$N$$ possible balls to be drawn. In order to better understand, what is happening in the experiment, imagine that the different symbols of an alphabet are written on the $$N$$ balls in order to distinguish them. In this case, the outcome of such an experiment is a word from this alphabet, and there are exactly

$|\Omega|=N^n$

possible words (outcomes of drawing $$n$$ letters from an alphabet of $$N$$ symbols). Now, among $$n$$ such letters, the binomial coefficient $$\binom nk$$ gives us the number of ways of writing $$k$$ letters in black ink (for a black ball drawn) and $$n-k$$ letters in white ink (for a white ball drawn). In addition, the number of words of length $$k$$, which can be written in black ink from an alphabet with $$M$$ symbols is $$M^k$$ and the number of words of length $$n-k$$, which can be written in white ink from an alphabet with $$N-M$$ symbols is $$(N-M)^{n-k}$$. Thus, according to the fundamental counting principle, the number of outcomes of the event

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

is $|A|=\binom nk (M)^k(N-M)^{n-k}.$

It follows from $$( * )$$ that

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

Github: ### References

#### Bibliography

1. Bosch, Karl: "Elementare Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition