Tourney Seeds Extended

8 top seeds, numbered 1 to 8 enter the quarter-final of a tourney. Seed number 1 is the top ranked seed, followed by seed number 2, then seed number 3, and so on...

  • Pairing in both the quarter-final and the semi-final are made randomly among the players entering that round.

  • A higher seeded player defeats the immediate next lower seeded player with a probability of 75%, and always defeats all other lower seeded players.

The probability that seed number 1 reaches the final of the tourney can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Find \(a +b\).


As an explicit example, this is how the seeds would perform during a match:

  • Seed number 1 defeats seed number 2 with 75% chance and always defeats seed numbers 3-8.

  • Seed number 2 defeats seed number 1 with 25% chance, defeats seed number 3 with 75% chance and always defeats seed numbers 4-8.

  • Seed number 3 always loses to seed number 1, defeats seed number 2 with 25% chance, defeats seed number 4 with 75% chance, and always defeats seed numbers 5-8.

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