Hypergeometric — the SuperEnalotto

Draw-based lotteries are a natural application of the hypergeometric: numbers are drawn without replacement from a finite set.

Example — The SuperEnalotto

You draw $6$ numbers out of $90$ without replacement. The probability of guessing exactly $k$ of them having played $6$ numbers is hypergeometric with $N=90$, $K=6$ (the $6$ winning ones), $n=6$ (played), $k=$ number of hits: $P(\text{Sei}) = P(X=6) = \frac{\binom{6}{6}\binom{84}{0}}{\binom{90}{6}} = \frac{1}{622\,614\,630}\approx 1{,}6\cdot 10^{-9}.$

About one chance in six hundred and twenty-two million: it is the number of ways of choosing $6$ numbers out of $90$. The hypergeometric model makes it immediate to compute also the probability of a “5”, a “4” and so on, simply by changing the value of $k$.

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Topics: Probability distributions
Concepts: Hypergeometric distribution
Methods: Hypergeometric distribution
Skills: Probability calculation · Modelling