Ten distinct numbers are arranged in a uniformly random order. You are told only that the number in position 1 is larger than the number in position 2. What is the probability that position 1 holds the largest of all ten?

Ten distinct numbers are arranged in a uniformly random order. You are told only that the number in position 1 is larger than the number in position 2. What is the probability that position 1 holds the largest of all ten?

Approach: Both the event you are told and the event you are asked about are symmetry statements about a uniform arrangement. Compute each unconditional probability and divide.

1/5. The arrangement is a uniform random permutation, so position 1 holds the maximum with probability 1/10, and when it does it certainly beats position 2. Hence P(maximum at position 1 and position 1 > position 2) = 1/10. The conditioning event has probability 1/2 by a symmetry argument between the two positions, so the conditional probability is (1/10)/(1/2) = 1/5. In general the answer is 2/n, so winning a single pairwise comparison exactly doubles the chance of being the global maximum, and it is the only information a one-off comparison carries.

Follow-up: Given the same information, what is the expected rank of the number in position 1?

Key concepts: conditional probability, random permutation, symmetry argument, pairwise comparison.