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The Truel
---------
Mr. Black, Mr. Gray, and Mr. White are fighting
in a truel. They each get a gun and take turns
shooting at each other until only one person is
left. Mr. Black, who hits his shot 1/3 of the
time, gets to shoot first. Mr. Gray, who hits his
shot 2/3 of the time, gets to shoot next, assuming
he is still alive. Mr. White, who hits his shot
all the time, shoots next, assuming he is also
alive. The cycle repeats. If you are Mr. Black,
where should you shoot first for the highest chance
of survival?
He should shoot at the ground.
If Mr. Black shoots the ground, it is Mr. Gray's
turn. Mr. Gray would rather shoot at Mr. White
than Mr. Black, because he is better. If Mr.
Gray kills Mr. White, it is just Mr. Black and
Mr. Gray left, giving Mr. Black a fair chance of
winning. If Mr. Gray does not kill Mr. White, it
is Mr. White's turn. He would rather shoot at Mr.
Gray and will definitely kill him. Even though
it is now Mr. Black against Mr. White, Mr. Black
has a better chance of winning than before.