Will the two friends get paired up?

Question

Eight players are randomly split into 4 pairs for a doubles tournament. Two of the players, Ana and Ben, are close friends. What is the probability that Ana and Ben are paired together?

Accepted Answer

The cleanest view: focus on Ana. Whatever the pairing procedure, Ana ends up partnered with exactly one of the other 7 players, and by symmetry each of those 7 is equally likely to be her partner. Ben is one of those 7 players, so P(	ext{Ana paired with Ben}) = \frac{1}{7} \approx 0.143. You can confirm with counting: the number of ways to pair up 8 people is \frac{8!}{2^4\,4!} = 105. Pairing Ana with Ben leaves 6 people to pair, in \frac{6!}{2^3\,3!} = 15 ways. Then \frac{15}{105} = \frac{1}{7}. Takeaway. For "do two specific items land together," fix one item and ask which slot the other f…

Will the two friends get paired up?

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