Answer to Puzzle #54: Two Bullet Russian Roulette

54. We are to play a version of Russian Roulette, the revolver is a standard six shooter but I will put two bullets in the gun in consecutive chambers. I spin the chambers, put the gun to my head pull the trigger and survive. I hand you the gun and give you a choice...
You may put the gun straight to your head and pull the trigger, or you may re-spin the gun before you do the same.

What is your choice and why? How does this differ from the case with only one bullet?

I found this puzzle whilst googling Wall Street interview questions. It's good fun so lets see what we can do with it:

picture of a revolver containing two bullets for russian rouletteBefore reading the answer can I interest you in a clue?

If you have never thought about the analogous question with 1 bullet, (and why would you?) we can quickly cover that now. One bullet can be in 6 positions as shown.
1 B-----
2 -B----
3 --B---
4 ---B--
5 ----B-
6 -----B
Obviously you would, with a randomised gun, have a 1 in 6 chance of dying. However, if someone has just fired and survived then we have eliminated the first scenario, B-----, it definitely wasn't that. And we are on the second chamber. We have essentially reduced the game to the following (removing the first row, because it wasn't that, and the first column because that chamber has been fired.):
2 B----
3 -B---
4 --B--
5 ---B-
6 ----B
Which is just the same as a standard game of Russian Roulette if there were only 5 chambers in the gun. A 1 in 5 chance of dying is worse than a 1 in 6. So in the case of one bullet it would be in your interest to spin and re-randomise the gun.

Back to Two Bullets

We will apply the same technique as above but first I would like to explain it intuitively. There's more than a hint of the game battleships here. As the second player we can only be hit by the second bullet in the chamber, remember they are consecutive, if the first bullet has already been fired, and we know it has not. By not spinning we have essentially removed an entire bullet from our problem. That is why we should fire the gun as is. Now lets model it.

If we randomise the gun we have a 2 in 6 or 1 in 3 chance of dying. The possible set ups at the start of the game are as follows:
1 BB----
2 -BB---
3 --BB--
4 ---BB-
5 ----BB
6 B----B
As player 1 didn't die we know it was not scenarios 1 and 6, eliminating those rows, and the first column as that chamber is no longer in the game - we're left with this:
2 BB---
3 -BB--
4 --BB-
5 ---BB
Giving us a 1 in 4 chance of dying. By re spinning the gun it would have been 1 in 3. So in the case of 2 adjacent bullets it would not be in your interests to re-spin the chamber.

Assumptions

A fundamental assumption is that your objective is to survive the game. After that we're down to the usual, that the gun is randomised and not influenced by the weight of the bullets. The gun never gets jammed, you never miss. All very reasonable perhaps so implicit that you wouldn't think to mention them.





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