Top two seeds meeting in a 128-player final

Question

A knockout chess event has $128$ entrants with distinct ratings and a random bracket. The higher-rated player in any game always wins and advances. What is the probability that the two highest-rated players face each other in the final?

Short answer

$64/127$ . The top player always reaches the final, and the second-best reaches it unless they meet earlier, i.e. exactly when they start in opposite halves. Of the $127$ other slots, $64$ are in the other half.

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Lessons behind this question

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

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