Pairs left in the drawer

HardCombinatorics Expected Value~7m

A drawer holds $6$ distinct pairs of socks ( $12$ socks). You pull out $8$ socks uniformly at random. What is the expected number of complete pairs that remain in the drawer (among the $4$ socks left behind)?

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$8$ drawn leaves $4$ behind. Use linearity: $\mathbb{E}=\sum_{\text{pairs}}\mathbb{P}(\text{pair fully remains})$ .
A given pair remains iff both its socks are in the $4$ left: $\binom{10}{2}/\binom{12}{4}=1/11$ ; times $6$ .

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0.5455 (± 0.001)

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Pairs left in the drawer

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