Top two seeds meeting in a 128-player knockout final

Question

A single-elimination chess event has $128$ players with distinct ratings, placed in the bracket uniformly at random. The higher-rated player always wins. What is the probability that the two highest-rated players face each other in the final?

Short answer

$64/127$ . The two best players meet in the final exactly when they start in opposite halves of the bracket. Given the top player's slot, $64$ of the other $127$ slots lie in the other half.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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