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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=52 marbles with m=20 red ones.

 Probability Format Numeric (between 0 and 1) Percentage (in %) Fraction A/B (exact or approximate) Odds (About ≈) X in N Chance

## Probabilities for multiple Draws

Example: Calculation of the probability of having drawn the card A♠ at least once, after 100 repeated drawings (with replacement) in a 52-card deck.

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

 Probability Format Numeric (between 0 and 1) Percentage (in %) Fraction A/B (exact or approximate) Odds (About ≈) X in N Chance

## Draws Simulator

⮞ Go to: Random Selection

### What is a picking probability? (Definition)

In combinatorics, random draws make it possible to evaluate the statistical probabilities of selecting a subset of objects (marbles, cards, etc.) from a total set.

Mathematical models make it possible to predict the distribution of draws without having to carry them out.

Simulation is not necessary, mathematical formulas give exact results.

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

### How to compute a probability based on the previous draw?

For a draw with replacement, the previous and following draws are completely independent. This may seem counter-intuitive, and it is also a classic mistake that casino or lotto players make, but in no case does the fact that an element has been drawn during a previous draw increase or decreases his chances of being drawn in the next draw.

For a draw without replacement, on the other hand, the elements drawn are to be removed from the following draws, so the probability must take this change into account.

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