Section 7.2

Sampling Without Replacement

Hypergeometric and negative-hypergeometric probabilities: the k-th success on a given draw, the most likely stopping time, each player receiving one ace.

Toolkit

Without replacement, order the $n$ items into a uniformly random permutation of the $N$ slots; any subset of positions is equally likely, so probabilities reduce to counting arrangements rather than simulating draws sequentially.

Key facts:

Hypergeometric: drawing $n$ from $N$ with $K$ special items, $\mathbb{P}(X=k)=\binom{K}{k}\binom{N-K}{n-k}/\binom{N}{n}$ .
Negative hypergeometric: probability the $r$ -th special item appears exactly on draw $m$ is

\mathbb{P}(T_r=m)=\frac{\binom{m-1}{r-1}\binom{N-m}{K-r}}{\binom{N}{K}},\qquad r\le m\le N-K+r,

more simply $\mathbb{P}(T_r=m)=\dfrac{\binom{m-1}{r-1}\binom{N-m}{K-r}}{\binom{N}{K}}$ .

Mode of such a pmf: compare consecutive terms via the ratio $\mathbb{P}(T_r=m+1)/\mathbb{P}(T_r=m)$ and find where it crosses $1$ .
Sequential placement trick: each new special item independently lands among the remaining $N-j$ slots, giving product formulas like $(39/51)(26/50)(13/49)$ for distributing aces across piles.

Problems