Lotto, the five-number win

Question

Lotto draw on a wheel: $5$ numbers drawn out of $90$. What is the probability of guessing the five-number win (cinquina) ($5$ numbers played, all winning)?

Solution

Hypergeometric model with $N=90$, $K=5$ (the five drawn), $n=5$ (the numbers played), $k=5$: $P(X=5) = \frac{\binom{5}{5}\binom{85}{0}}{\binom{90}{5}} = \frac{1}{\binom{90}{5}} = \frac{1}{43\,949\,268} \approx \boxed{2{,}3\cdot 10^{-8}}.$ About one chance in $44$ million: the five-number win is the most improbable prize in the game of Lotto.

Links

Topics: Probability distributions
Concepts: Hypergeometric distribution
Methods: Hypergeometric distribution
Skills: Combinatorial calculation · Probability calculation
Exercise type: Probability problem