Roulette with a respin before every trigger pull

Two players take turns with a six-chamber revolver containing one bullet, but now the cylinder is spun again before every single pull. Should you go first or second, and what is your probability of being shot under each choice?

Go second: you are shot with probability $5/11$, versus $6/11$ going first. Each pull fires with probability $\tfrac16$, independently. The first player is shot with probability $\tfrac16\left(1+\left(\tfrac56\right)^2+\left(\tfrac56\right)^4+\dots\right)=\dfrac{1/6}{11/36}=\tfrac6{11}$.