Variance of the heart count in an eight-card hand

Question

Eight cards are dealt from a well-shuffled standard deck. What is the variance of the number of hearts among them?

Short answer

$\tfrac{22}{17}$ . The heart count is hypergeometric. Its variance is the binomial value $8\cdot\tfrac14\cdot\tfrac34=\tfrac32$ times the finite-population correction $\tfrac{52-8}{52-1}=\tfrac{44}{51}$ , giving $\tfrac{22}{17}$ . (Indicator covariances $-\tfrac{1}{272}$ per pair give the same.)

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Lessons behind this question

Covariance and Correlation
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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