Does one group own both extremes?

Question

Group A and Group B each contribute 4 independent draws from the same continuous distribution, for 8 in total. No two values tie.

What is the probability that both the overall maximum and the overall minimum come from Group A?

Accepted Answer

Pool all 8 values, which are exchangeable. First, the maximum belongs to Group A with probability \frac{4}{8}. Given that, one of Group A's four values is used up as the maximum, leaving 3 Group A values among the remaining 7. The minimum is then equally likely to be any of those 7, so it is a Group A value with probability \frac{3}{7}. Multiplying: P(	ext{max and min both in A}) = \frac{4}{8}\cdot\frac{3}{7} = \frac{12}{56} = \frac{3}{14} \approx 0.214. Takeaway. For two extremes at once, place them one after another: the second extreme draws from the group's remaining share of the shrunken…

Does one group own both extremes?

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