Russian Roulette — No Re-Spinning

A 6-chamber revolver has its chambers arranged in a circle, numbered 1 through 6. A single bullet is placed into one of the 6 chambers. The barrel is then randomly spun so that each chamber is equally likely to be under the hammer. Two players take turns pulling the trigger — with the gun pointed at their own head — without further spinning. After each pull, the barrel advances to the next chamber. The person who gets killed loses.

If you are one of the players and can choose to go first or second, how will you choose? What is your probability of losing?