Two Bullets, One Spin

A revolver has six chambers. Two bullets are loaded into two adjacent chambers, the cylinder is spun once, and the trigger is pulled. It clicks on an empty chamber. The cylinder is not reloaded. For the next pull, are you safer spinning the cylinder again first, or pulling straight away?

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#Spinning again

A fresh spin erases the past. Two of the six chambers hold a bullet, so

\PP(\text{fires} \mid \text{spin}) = \tfrac{2}{6} = \tfrac13. \tag{1}

#Pulling without spinning

Number the chambers $1$ through $6$ around the cylinder with the bullets in the adjacent chambers $1$ and $2$ , so the empties are $3, 4, 5, 6$ and each pull advances by one. The first pull was empty, so it sat on one of $3, 4, 5, 6$ , all equally likely. The next chamber holds a bullet only when the current one is $6$ , which advances to $1$ . That is just $1$ of the $4$ empty positions,

\PP(\text{fires} \mid \text{no spin}) = \tfrac14. \tag{2}

Bullets sit in chambers 1 and 2, leaving 3 through 6 empty, and each pull steps forward one chamber. After an empty click you are equally likely to be on any of the four empties, but only chamber 6 steps into a bullet, so pulling again fires with probability one quarter against one third for a fresh spin.

#Result

Since $\tfrac14 < \tfrac13$ , do not spin. Surviving one empty chamber means you are probably deep in the run of four empties, and only the single empty just before the two bullets leads into danger.