Question: There are 15 balls in a container, of which 10 are blue and 5 are red. 9 balls are drawn at random from the container now without replacement. What is the probability of finding exactly 4 blue balls in this sample?
A population of N elements is given; M of which have a certain property. You take a sample of size n (without putting it back). Then the random variable X = number of elements in the sample with the desired property can take at most the values k = 0,1,2,…,n. The probability of finding exactly k elements with the desired property in the sample is:
(For k > M respectively n - k > N - M is P(X = k) = 0) according to the definition of the binomial coefficient – we cannot draw more elements with a certain property than there are available). X is hypergeometrically distributed and its corresponding probability distribution is a hypergeometrical distribution with the parameters n, M and N.
Properties of the hypergeometric distribution
Expected value: μ = E(X) =
Variance: σ² = Var(X) =
Further information can be found in "Teschl Mathematik für Informatiker", Volume 2, Edition 3, Chapter 28.1, Page 303-306.
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