The point is that once the order of firing has been established and the chamber has been spun, the person who is going to die has been absolutely and immutably decided. It is not decided when the bullet is fired.
If the bullet is ballanced by equal weight slugs in the other five chambers, your chance of dying is 1/6 and that's that. I you want maths, here goes (assuming that you don't have to carry on pulling the trigger once the bullet has been fired):
First person, Chance of having to pull trigger 1/1; chance of getting bullet 1/6; overall chance of death 1/6.
Second person, Chance of having to pull trigger 5/6; chance of getting bullet 1/5; overall chance of death 1/6.
Third person, Chance of having to pull trigger 4/6; chance of getting bullet 1/4; overall chance of death 1/6.
Fourth person, Chance of having to pull trigger 3/6; chance of getting bullet 1/3; overall chance of death 1/6.
Fifth person, Chance of having to pull trigger 2/6; chance of getting bullet 1/2; overall chance of death 1/6.
Last person, Chance of having to pull trigger 1/6; chance of getting bullet 1/1; overall chance of death 1/6.
But remember, these probablilities only actually exist up to the point that the chamber is spun and the fireing order is decided. After that point, the probability of death for one participant is 1 and for the other five it's zero, they just don't know who.