The Gunman's Offer

Probability Level 2

You are a prisoner about to be executed by a gunman. As you move into position the gunman takes a six-chamber revolver and places inside two bullets next to one another. He spins the chamber, locks it in place, aims and fires...and nothing happens. At this point he tells you he will fire one more time and if you are still standing afterwards you are free to go. He gives you two options- either fire again, or spin the chamber and then fire.

Which option provides you the best chance of walking away unscathed?

Spin the chamber and then fire again It doesn't matter Fire again without spinning the chamber

4 solutions

John Penn
Oct 16, 2015

If you elect for the gunman to spin the chamber before firing a second time then you essentially have a 2/3 chance of survival. Of the six chambers, two have bullets, which means you have a 2/6 chance of being shot and a 4/6 of not. Now, if you elect for the gunman to fire again, then you have a 3/4 chance of surviving. You know that the two bullets are in sequential order, which means that the second bullet cannot be fired before the first. Therefore, since you know that the first shot was empty, there are three more potential empty shots as well as the first bullet- 75% chance of survival.

What I thought was: First shot you have 4/6 chance of surviving. Only in one of the chambers, that made you survive, the next shot will have a bullet, because the bullets are NEXT one to another. As we have 4 free chambers, we have 3/4 chance of surviving. So you have 4/6*3/4 chance of surviving, or 1/2 = 50%. When you spin the chamber, you have 4/6 chance of surviving, or 2/3 = 66%. So it's better to spin... What's wrong with my logic?

Felipe Magalhães - 5 years, 3 months ago

You already shot the first bullet so you dont need to count the chance to survive the first bullet, so its not 3/4*2/6 (50%) its just 3/4 = 75% against 2/6 = 66.6666%

André Sanchez de Toledo - 5 years, 3 months ago

Against 4/6 = 66.666%

Senthil Kumar - 5 years, 3 months ago

Norman Rowe
Feb 23, 2016

Just fire. The two bullets are together. There is no way the next shot will be the second bullet since the last shot was not a bullet. There is only a 1 in 6 chance that the next shot is a bullet. In other word since the last shot did not have a bullet, then you will only be shot with a bullet on the next fire if the previous no bullet shot was the last empty chamber before the 2 bullets.

Never mind. 1 in 4 not 1 in 6.

Norman Rowe - 5 years, 3 months ago

actually it could be the next bullet because if the empty chamber that was shot was the chamber JUST befor the loaded ones then you will be shot

Steven Butler - 5 years, 3 months ago

Weiting Hong
Feb 24, 2016

This looks like a monty hall problem. After that shot, you know that there are still two bullets and 3 empty spots left. If you request the gunman to fire again without spinning, that means he has to fire in sequential order. Since second bullet cannot come before first and he can only shoot one more time, the second bullet is out of consideration. This means we now have a 1/4 chance of being shot and 3/4 chance of survival. Whereas if we request another spinning, our probability falls back to 1/3 chance of being shot and 2/3 chance of suvival. Therefore, in comparison, it is better to request the gunman to fire again without spinning the chamber.

Lance Fernando
Feb 24, 2016

Quite easy, if in his first try he wan't able to fire his gun, perhaps it's because there's no bullet. Ask him to fire once more - you know he can't fire anymore once he didn't fire it in the first place.

The Gunman's Offer

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