Two bullets are loaded into a gun’s round barrel consecutively.

The barrel has a capacity of 6.

The gun is fired once, but no bullet is shot.

So does rolling the barrel (shuffling) before next shot increase the probability of firing a bullet?

Once again, this is a good moment to pause the article and give the problem a go yourself. When you’re ready, keep reading for the solution. And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

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Solution

Initial Misstep: If the two bullets are randomly put instead of consecutively, then, after firing one empty shot, there are $2$ bullets and $5$ total slots.

The probability would be $2\mathrm{/}5=40\mathrm{\%}$, but that’s not the case here.

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In fact the correct probability is 1/4 = 25% because the bullets are put consecutively.

The easiest way to see it is to list all possibilities.

Suppose the bullets occupy consecutive chambers 1 and 2.

1 2 3 4 5 6
B B E E E E

You fire once, and nothing happens. Therefore, the firing pin cannot have started on chambers 1 or 2. It must have been on one of the empty chambers:

Possible first position of firing pin:

1 2 3 4 5 6
B B X E E E

1 2 3 4 5 6
B B E X E E

1 2 3 4 5 6
B B E E X E

1 2 3 4 5 6
B B E E E X

These four cases are equally likely.

Now suppose the cylinder advances clockwise after the empty click.

Case 1: Started on chamber 3

Before:
B B [E] E E E

After advancing:
B [B] E E E E

Next shot: BULLET ✓

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Case 2: Started on chamber 4

Before:
B B E [E] E E

After advancing:
B B [E] E E E

Next shot: EMPTY ✗

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Case 3: Started on chamber 5

Before:
B B E E [E] E

After advancing:
B B E [E] E E

Next shot: EMPTY ✗

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Case 4: Started on chamber 6

Before:
B B E E E [E]

After advancing:
B B E E [E] E

Next shot: EMPTY ✗

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There are 4 equally likely possibilities, and only 1 leads to a bullet.

First empty click    Next chamber
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3                     Bullet ✓
4                     Empty
5                     Empty
6                     Empty

Therefore,

If you spin the cylinder instead, all this information is lost. The cylinder is reset to a random orientation, so the probability becomes

This is the counterintuitive part of the puzzle: not spinning is actually safer because the empty click gives you information about where the consecutive bullets cannot be.

And that’s our answer. How amazing. Hey, don’t forget to clap👏 the article as a token of appreciation. Thank you 🦁