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Picking Probabilities

Tool to make probabilities on picking/drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement.

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Picking Probabilities -

Tag(s) : Combinatorics

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Picking Probabilities

Probabilities for a Draw without Replacement

Example: Probability to pick a set of n=10 marbles with k=3 red ones (so 7 are not red) in a bag containing an initial total of N=100 marbles with m=20 red ones.









Probabilities for multiple Draws

Example: Calculation of the probability of having drawn the number '23' after 200 drawings of a 50-face dice.









Answers to Questions (FAQ)

How to compute a probability of picking without replacement?

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.

How to compute a probability of picking with replacement?

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 $$ \sum_{i=0}^N (-1)^{N-i}{\binom{N}{i}}\left(\frac{i}{N}\right)^n $$

Source code

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