Which of three groups holds the maximum?

Question

Three pools contribute independent draws from the same continuous distribution: pool P has 2 values, pool Q has 3 values, and pool R has 5 values, for 10 in total. No two values tie.

What is the probability that the overall largest value comes from pool R (the biggest pool)?

Accepted Answer

Pool all 10 values. They are exchangeable, so each is equally likely to be the largest. Pool R owns 5 of the 10 values, so the maximum comes from R with probability \frac{5}{10} = \frac{1}{2} = 0.5. The same reasoning gives \frac{2}{10} for pool P and \frac{3}{10} for pool Q, and the three shares sum to 1. Takeaway. With any number of groups, the maximum lands in each group with probability equal to that group's proportion of the pool.

Which of three groups holds the maximum?

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