At least one couple matched in random pairing

Question

Six people consisting of three couples are split uniformly at random into three pairs. What is the probability that at least one couple ends up paired together?

Short answer

$7/15$ . Use inclusion–exclusion over the three couples. A given couple is paired with probability $\tfrac15$ ; any two given couples are both paired with probability $\tfrac15\cdot\tfrac13$ , and that automatically pairs the third. Hence $\tfrac35-\tfrac{3}{15}+\tfrac{1}{15}=\tfrac{9-3+1}{15}=\tfrac{7}{15}$ .

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

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Counting and Combinatorics
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

Start with a free lesson: Probability Fundamentals is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free