Triple Duel

Phoenix Math Tutoring Cowboy Problem

Three Cowboys Arden, Begley, and Corry compete in a duel. Arden and Begley are both sure shots (they hit their targets 100 percent of the time) while Corry only hits his target 50% of the time. Everybody has perfect knowledge (knows everybody’s percents, knows that everbody knows everybody’s pecents, knows everybody knows that everybody knows everybody’s percents, etc…).

They draw lots to determine the shooting order. Who has the best chance of survival?


Even though Corry has only a 50 percent chance of hitting his opponent, he actually has the best chance of surviving, at 50 percent. When drawing lots there are three situations that may arise:

  • A shoots first: he will shoot B, and then C will have a 50% chance. A/C have a 50/50 split.
  • B shoots first: same as above. B/C have a 50/50 split.
  • C shoots first: C misses intentionally. Depending on who shoots second, A or B shoots the other, and then C has a 50 percent chance.  Thus A/B/C is 25/25/50.

Overall A has a 25% chance of survival, B has a 25% chance, and C has a 50% chance.

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