Spouses landing together in a random pairing

Question

Eighteen people, made up of $9$ married couples, are split into $9$ pairs uniformly at random (pairs may be any two people, regardless of gender). On average, how many of the pairs are original married couples?

Short answer

$9/17$ . Use an indicator for each couple. A given person is equally likely to be paired with any of the other $17$ people, so each couple is reunited with probability $\tfrac1{17}$ , and linearity gives $9\cdot\tfrac1{17}$ .

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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