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Picking Probabilities

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

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Picking Probabilities -

Tag(s) : Combinatorics,Mathematics

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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.

 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, etc.) in a box (bag, drawer, etc.) with and without replacement is a common exercise in probability.

## Answers to Questions

### How to compute a probability of picking without replacement?

Consider a set of N objects among which m are different. 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:

$$p(X=k)=\frac{C_{m}^kC_{N-m}^{n-k}}{C_N^n}$$

C represents the combination operator.

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.

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 having picked all N objects after n draws is given by the formula

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

C represents the combination operator.

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## Questions / Comments

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