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Omni-MATH / Ash and Gary independently come up with their own lineups of 15 fire, grass, and…

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problem

Ash and Gary independently come up with their own lineups of 15 fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team's lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary's lineup, Ash chooses a lineup that maximizes the probability pp that his monsters are the last ones standing. Compute pp.
Plain-text mathematical notation (without MathML)
Ash and Gary independently come up with their own lineups of 15 fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team's lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary's lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute p.
Original LaTeX notation
Ash and Gary independently come up with their own lineups of 15 fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team's lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary's lineup, Ash chooses a lineup that maximizes the probability $p$ that his monsters are the last ones standing. Compute $p$.

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Source and history

task_id
b9c5ee24-8f77-5101-b61c-825dfdc685df
task_key
test--b9c5ee24-8f77-5101-b61c-825dfdc685df
task_revision_id
1
config / split
/ test

Official source

initial import