Probability puzzles.

Problem. 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?

https://brainstellar.com/puzzles/easy/1

Solution.

Let the holes in the barrel be $1,2,3,4,5,6$ labelled clockwise. WLOG, suppose the firing pin was on hole $1$. Also, suppose the barrel rotates clockwise. It’s given that the gun is fired once, but no bullet was shot. It implies that the bullets must have been loaded in $(2,3)$, $(3,4),(4,5)$ or $(5,6)$. There’s a $1/4=25$ percent chance of firing a bullet, on the next shot; that is, if the bullets were loaded in $(5,6)$.

On the other hand, if the barrel is shuffled, there are $6$ holes the firing pin could point to; out of which $2$ are favorable. That’s a $2/6\approx 33$ percent chance of firing the bullet.

So, shuffling the barrel increases the chances of firing a bullet from $25$ percent to $33$ percent.