Tool to make probabilities on picking objects. Calculation of probabilities of drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement is a common exercise in probability.

Picking Probabilities - dCode

Tag(s) : Combinatorics

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Tool to make probabilities on picking objects. Calculation of probabilities of drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement is a common exercise in probability.

For a set of $ N $ objects among which $ m $ are different (distinguishable). The **probability of drawing** a total of $ n $ objects and that among these $ n $ objects there are $ k $ objects that are part of the $ m $ different ones, is given by a hypergeometric distribution: $$ p(X=k)=\frac{C_{m}^kC_{N-m}^{n-k}}{C_N^n} = \frac{ \binom{m}{k} \binom{N-m}{n-k} }{ \binom{N}{n} } $$

C represents the combination operator.

__Example:__ Probability to draw $ k=5 $ red card among the $ m=26 $ red cards in a deck of $ N=52 $ cards by drawing $ n=5 $ cards.

__Example:__ Probability to draw all $ k=3 $ black ball in a bowl with $ N=25 $ balls among which $ m=3 $ are black, by picking $ n=3 $ balls.

The probability of never having picked a given item among $ N $ objects after $ n $ random draws is given by the formula $$ \left(1-\frac{1}{N}\right)^n $$

The probability of having picked at least once a given item among $ N $ objects after $ n $ random draws is given by the formula $$ 1-\left(1-\frac{1}{N}\right)^n $$

The probability of having picked all $ N $ objects (discernible or indistinguishable) after $ n $ random draws is given by the formula $$ \sum_{i=0}^N (-1)^{N-i}{\binom{N}{i}}\left(\frac{i}{N}\right)^n $$

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