Answer: choose to go second, you have a 4/6 or 2/3 chance of winning.

Let us label the chambers for our convenience as C1, C2... C6.

Now, when the cylinder is spun, there can be the following six outcomes.

1. If C1 is fired first: Player 1 dies.

2. If C2 is fired first: Player 1 dies.

3. If C3 is fired first: Player 1 dies.

4. If C4 is fired first: Player 2 dies (First shot, player 1, C4 empty. Second shot player 2, C5, empty. Third shot player 1, C6 empty. Fourth shot player 2, C1 not empty.)

5. C5 is fired first: Player 1 dies (First shot, player 1, C5 empty. Second shot player 2, C6, empty. Third shot player 1, C1 not empty.)

6. C6 is fired first: Player 2 dies (First shot, player 1, C6 empty. Second shot, player 2, C1, not empty)

Therefore, if you choose to go second, you have a 4/6 or 2/3 chance of winning.