Hypergeometric: Sampling

Probability · Axiom Academy

EXAMPLE Hypergeometric Distribution: Sampling Draw 5 cards from a standard deck without replacement. Find P(exactly 2 hearts) and compare to the binomial approximation. Excellent work! You've completed this hypergeometric distribution example. Here's what we learned: Hypergeometric distribution: Models sampling WITHOUT replacement from a finite population. Key difference from binomial: probabilities change as we draw. Formula structure: P(X = k) = [C(K,k) × C(N-K, n-k)] / C(N,n) counts favorable outcomes divided by total outcomes. Parameters: N = population size, K = number of "success" items in population, n = sample size, k = desired successes in sample. Binomial approximation: When N is large relative to n (typically n/N < 0.05), hypergeometric ≈ binomial. Here 5/52 ≈ 0.096, so approximation is decent but not perfect. Practical difference: Hypergeometric (27.4%) vs Binomial (26.4%) differ by about 1% in this case. Use hypergeometric when sampling without replacement from small populations. For large populations, binomial is a good approximation!

This is the written version of the interactive lesson above. See the full Probability course.