Picking Probabilities

In combinatorics, random draws are used to study the probabilities of selecting a subset of objects, such as marbles or cards, from a larger set.

Thanks to mathematical models, it is possible to predict how these draws will be distributed without actually performing them. Therefore, there is no need for simulation: the formulas provide precise and accurate results.

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.

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

In a draw with replacement, each draw is completely independent of the others. In other words, the fact that an element has been drawn before has no influence on the probability of it being drawn again. This may seem surprising at first — it's a common mistake among lottery and casino players — but the odds remain exactly the same for each draw.

In contrast, in a draw without replacement, things change: once an element is drawn, it is no longer available for subsequent draws. The probability therefore evolves gradually, as the total number of elements decreases.