Search for a tool
Picking Probabilities

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.

Results

Picking Probabilities -

Tag(s) : Combinatorics

Share
dCode and you

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!

Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Picking Probabilities tool. Thank you.

# 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.

 Probability to draw at least 1 time a given item not even once (0 time) a given item

## Probabilities for a Draw with Replacement

Example : Probability to pick at least once each card from a deck of N=50 cards after n=200 drawings.

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.

### 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 source code of the script Picking Probabilities online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Picking Probabilities script for offline use on PC, iPhone or Android, ask for price quote on contact page !