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Given two urns with an identical set of $n$ balls of $n$ different colors, where $n$ is even. After drawing without replacement $n/2$ pairs from the first urn and also drawing without replacement $n/2$ pairs from the second urn, what is the probability that at least one identical pair has been drawn? The order within a pair should not matter, so (A,B) is considered identical to (B,A).

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  • $\begingroup$ Welcome to Stats.SE. Take the opportunity to take the tour( stats.stackexchange.com/tour), if you haven't done it already. See also some tips on how to ask, on formatting help and on writing down equations using LaTeX / MathJax. If your question is clear and focused on your specific difficulty and you show your effort in solving the problem, it's more likely to get good and helping answers. $\endgroup$ – Ertxiem - reinstate Monica Sep 11 at 17:41
  • $\begingroup$ What have you tried so far? Perhaps a similar problem is easier to solve: you may consider that each urn has $k$ identical balls. In this case, what is the probability of drawing one identical pair? $\endgroup$ – Ertxiem - reinstate Monica Sep 11 at 17:43
  • $\begingroup$ Hi! Welcome here... and I agree with @Ertxiem's suggestions about better describing what the difficulty is. Anyway, it's a bit difficult to understand the question: how many draws are done from the second urn? I would assume one, but the observation again without replacement makes me wonder whether it's a different number (e.g. $n/2$ like from the first urn, or anything in between). $\endgroup$ – polettix Sep 11 at 17:47
  • $\begingroup$ Are these pairs ordered or unordered? $\endgroup$ – whuber Sep 11 at 19:29
  • $\begingroup$ Thanks for the input, I rephrased the question. $\endgroup$ – gnebehay Sep 12 at 7:01

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