## Maximise Your Chances of Surviving Russian Roulette

Russian Roulette is a game played with a partially loaded gun which involves someone pointing a gun at your head and pulling the trigger. If there is no shell in that chamber, you survive. Suppose a gun is loaded with a cartridge and the barrel is spun. Your opponent points the gun, pulls the trigger, and there is no cartridge in that chamber. This means that for the moment you live.
The barrel rotates onto the next chamber. Now you have a choice. Do you ask for the barrel to be spun again, or do you allow the gun to be fired again from the current chamber? Obviously you want to maximise your chance of survival.
The gun has six chambers arranged in a circle. Suppose chamber 2 contained the cartridge. The arrangement is then:
Loaded, Empty, Empty, Empty, Empty, Empty.
The first chamber to be fired did not contain a cartridge so it could be chambers 2, 3, 4, 5, 6. If it was chamber 2, 3, 4, or 5 and the barrel is not spun again, you will not die when the gun is fired again, and if it was chamber 6 then you will die. The chance that you will not die from the second shot is 4/5, and the chance that you will die is 1/5..
Suppose now that the barrel is spun again so a chamber is randomly selected. There is one cartridge in in six chambers, so the probability of selecting a loaded chamber is 1/6.
You are more likely to survive if you ask for the barrel to be spun again.
Suppose Chambers 1 and 2 contained the cartridges. The arrangement is then:
The first chamber to be fired did not contain a cartridge so it could be chambers 3, 4, 5 or 6. If it was chamber 3, 4, or 5 and the barrel is not spun again, you will die when the gun is fired again, so the chance that you will die from the second shot is 1/4.
Suppose now that the barrel is spun again so a chamber is randomly selected. There are two cartridges in six chambers, so the probability of selecting a loaded chamber is 1/3.
You are more likely to survive if you do not ask for the barrel to be spun again.
Suppose Chambers 2 and 3 contained the cartridges. The arrangement is then:
The first chamber to be fired did not contain a cartridge so it could be chambers 1, 4, 5, 6. If it was chamber 6 and the barrel is not spun again, you will die when the gun is fired again, so the chance that you will die from the second shot is 1/4.
Suppose now that the barrel is spun again so a chamber is randomly selected. There are two cartridges in six chambers, so the probability of selecting a loaded chamber is 1/3.
You are more likely to survive if you do not ask for the barrel to be spun again.
Suppose Chambers 1, 2 and 3 contained the cartridges. The arrangement is then:
The first chamber to be fired did not contain a cartridge so it could be chambers 4, 5 or 6. If it was chamber 6 and the barrel is not spun again, you will die when the gun is fired again, so the chance that you will die from the second shot is 1/3.
Suppose now that the barrel is spun again so a chamber is randomly selected. There are three cartridges in six chambers, so the probability of selecting a loaded chamber is 1/2.
You are more likely to survive if you do not ask for the barrel to be spun again.
Suppose Chambers 1, 2, 3 and 4 contained the cartridges. The arrangement is then:
The first chamber to be fired did not contain a cartridge so it could be chambers 5 or 6. If it was chamber 6 and the barrel is not spun again, you will die when the gun is fired again, so the chance that you will die from the second shot is 1/2.
Suppose now that the barrel is spun again so a chamber is randomly selected. There are four cartridges in six chambers, so the probability of selecting a loaded chamber is 2/3.
You are more likely to survive if you do not ask for the barrel to be spun again.
Suppose Chambers 1, 2, 3, 4 and 6 contained the cartridges. The arrangement is then:
If you survive the first shot, then the barrel has rotated onto a chamber that contains a cartridge, so you should ask for the barrel to be spun again.
If all the chambers had cartridges, put a finger in the hole in your head to stop the bleeding.

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