Variance of the club count in an eight-card hand

Question

Deal $8$ cards from a well-shuffled $52$ -card deck. Let $X$ be how many of them are clubs. Find $\operatorname{Var}(X)$ .

Short answer

$22/17$ . $X$ is hypergeometric, so its variance is the binomial value $8\cdot\tfrac14\cdot\tfrac34=\tfrac32$ times the finite-population correction $\tfrac{52-8}{51}=\tfrac{44}{51}$ , giving $\tfrac{66}{51}=\tfrac{22}{17}$ .

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Lessons behind this question

Covariance and Correlation
Populations, Samples, and Sampling Distributions
Sample Spaces, Events, and the Probability Axioms

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