Picking Probabilities

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.

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