Hypergeometric vs binomial

Hypergeometric and binomial describe the same type of problem — counting the successes — but they differ on one point: whether or not there is replacement. The comparison clarifies when one approximates the other.

Property — Hypergeometric vs. binomial comparison

• Binomial $B(n,p)$: draws with replacement. The trials are independent, $p = K/N$ is fixed.
• Hypergeometric $\text{Hyp}(N,K,n)$: draws without replacement. The probability of success changes at each draw (it depends on how many balls remain).

The hypergeometric variance differs from the binomial one by the factor $\dfrac{N-n}{N-1}$, called the finite population correction factor: it equals $1$ when $N\to\infty$ (very large population $\Rightarrow$ replacement and no replacement become indistinguishable).

Property — Binomial approximation

If $N\to\infty$ with $K/N\to p$ fixed, the hypergeometric $\text{Hyp}(N,K,n)$ converges to the binomial $B(n,p)$: $\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}\ \xrightarrow{\,N\to\infty,\ K/N\to p\,}\ \binom{n}{k}p^k(1-p)^{n-k}.$ Rule of thumb: if $n/N < 0{,}1$ (sample less than $10\%$ of the population), the binomial approximation is very good.

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Topics: Probability distributions
Concepts: Binomial distribution · Hypergeometric distribution · Variance
Skills: Limit-case analysis · Probability calculation