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Probability the Two Strongest Teams Meet in a 16-Team Knockout Final

DRWQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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Calculate the probability that the two highest-ranked teams face each other in the championship match of a 16-team single-elimination tournament with probabilistic match outcomes.

Problem Description

A single-elimination tournament consists of 16 teams ranked strictly from 1 (the strongest) to 16 (the weakest), with no ties.

Before the tournament begins, the 16 teams are randomly assigned to the 16 bracket slots uniformly at random. The tournament structure is standard:

  • Round 1 (Round of 16): 8 matches, pairing adjacent slots (e.g., Slot 1 vs Slot 2, Slot 3 vs Slot 4, etc.).
  • Round 2 (Quarterfinals): 4 matches between winners of adjacent Round 1 matches.
  • Round 3 (Semifinals): 2 matches between winners of adjacent Round 2 matches.
  • Round 4 (Final): 1 match between the two semifinal winners.

In any match between two teams, the stronger team (the one with the smaller rank number) wins with probability p = 0.9, independent of all other matches and past outcomes.

Questions

  1. What is the exact probability that Team 1 and Team 2 meet in the Final?
  2. Express your answer both as an exact fraction in terms of p and as a decimal evaluated at p = 0.9.
  3. Show why the specific identities of the opponents faced by Team 1 and Team 2 prior to the final do not need to be tracked.

Hints

Hint 1

Consider the necessary and sufficient conditions regarding bracket placement for the two teams to even have the possibility of meeting in the final.

Hint 2

Once placed in the bracket, determine whether Team 2 can ever face an opponent stronger than itself prior to the final match.

Hint 3

Check whether the paths of Team 1 and Team 2 through their respective halves of the bracket are statistically independent conditional on their initial placements.

Follow-up questions

  • How does the result change if we ask for the probability that Team 1 and Team 3 meet in the final?
  • What is the probability of Team 1 and Team 2 meeting in the final under a standard seeded bracket (e.g., seed 1 and seed 2 placed on opposite ends) instead of random placement?
  • If the tournament had `2^k` teams instead of 16, what would the general formula be as a function of `k` and `p`?