Tool to make probabilities on picking/drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement.
Picking Probabilities - dCode
Tag(s) : Combinatorics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Picking Probabilities
Probabilities for a Draw without Replacement
Probabilities for multiple Draws
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
dCode retains ownership of the "Picking Probabilities" source code. Except explicit open source licence (indicated Creative Commons / free), the "Picking Probabilities" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Picking Probabilities" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Picking Probabilities" are not public, same for offline use on PC, tablet, iPhone or Android !
The copy-paste of the page "Picking Probabilities" or any of its results, is allowed as long as you cite the online source
Reminder : dCode is free to use.
