Pair among three socks from ten colour pairs

Question

A drawer holds $10$ pairs of socks, each pair a different colour. Three socks are pulled out at random. What is the probability that two of them share a colour?

Short answer

$\tfrac{3}{19}$ . Count the complement: three different colours can be chosen in $20\cdot18\cdot16$ ordered ways out of $20\cdot19\cdot18$ , i.e. $\tfrac{16}{19}$ . So a match occurs with probability $\tfrac{3}{19}$ .

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

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Lessons behind this question

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

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