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

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


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References

Bibliography

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