easy | probability |

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

Since the bullets are loaded consecutively, the next shot is also constrained.

Yes, shuffling increases the probability of firing a bullet from $25\%$ to $33.\bar{3}\%$)

**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 $2/5 = 40\%$, but that's not the case here.

**Correct step**: The probability of firing a bullet without a shuffle is $1/4 = 25\%$. To understand this, imagine that the firing pin was on one of the empty slots $(3, 4, 5, 6)$, and the first shot was taken, but no bullet was fired. Now assumming that the barrel rotates clockwise, the pin will move to one of these slots: $(2, 3, 4, 5)$. Out of these four slots, only the slot $(1)$ has a bullet. Hence probability of firing a bullet is $1/4 = 25\%$.
Note that the same is true in anti-clockwise direction.

After the shuffle, the state is reset. There are $6$ total slots with $2$ bullets, the probabilty of firing a bullet after a shuffle is $2/6 = 1/3 \approx 33\%$.

Thus, shuffling does increase the probability of firing a bullet (from $25\%$ to $33\%$)