Spin or not after a survived pull with adjacent bullets

A six-shot revolver has two bullets in adjacent chambers, and the cylinder is spun at random. Your opponent pulls the trigger first and survives. You may now either spin the cylinder again or fire the next chamber directly. By how much does your survival probability without spinning exceed your survival probability with spinning?

$\tfrac{1}{12}$. Spinning gives survival $\tfrac46=\tfrac23$. Without spinning, the opponent fired one of the four empty chambers, equally likely, and only one of those is followed by a bullet, so you survive with $\tfrac34$. The gap is $\tfrac34-\tfrac23=\tfrac1{12}$.