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Will the two best players meet in the final?

probabilitytournamentscounting

Eight players have strictly different strengths, and in any match the stronger player always wins. They are drawn at random into a standard knockout bracket: quarter-finals, semi-finals, final.

  1. What is the probability that the two strongest players meet in the final?
  2. What is the probability that the second-strongest player finishes as runner-up?
  3. Generalise to 2 to the power k players.
Show a hint

The strongest player wins everything. The second strongest reaches the final exactly when she is not in the same half of the bracket as the strongest player.

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