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Last modified: November 20, 2021 ✓

The Truel
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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.

 

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