Going first or second with one bullet and no respin

A six-chamber revolver holds a single bullet. The cylinder is spun once at the start and never again; two players then take turns pulling the trigger, each pull advancing to the next chamber, until the gun fires. Would you rather pull first or second?

It makes no difference. The bullet is equally likely to sit in any of the six chambers. The first player is shot if it is in chamber $1$, $3$ or $5$ and the second if it is in $2$, $4$ or $6$, so each faces probability $\tfrac12$.