Complete sock pairs left after drawing eight

Question

A drawer holds $12$ socks made up of $6$ different matching pairs. You grab $8$ socks at random, all subsets of size 8 being equally likely. On average, how many complete pairs are still in the drawer?

Short answer

$\tfrac{6}{11}$ . Use indicators: a given pair stays only if all 8 drawn socks come from the other 10, which has probability $\binom{10}{8}/\binom{12}{8}=\tfrac{45}{495}=\tfrac1{11}$ . By linearity, the expected number is $6\cdot\tfrac1{11}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Continuous Random Variables and Probability Density Functions

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