Wherease if the sixth person spins, then they have a 1/6 chance of blowing their brains out and a 1/6 chance the gun gets to them, for a total of only 1/36 chance of no brain, compared to a 6/36 chance of no brain for the first guy, 5/36 for the second guy (5/6 chance that he has to pull the trigger, 1/6 chance he dies if he has to pull the trigger), third guy has 4/6 chance of having to pull the trigger for a combined 4/36 (which reduce to 1/9), fourth guy has a fifty-fifty whether he has to pull the guy which gives him combined 3/36 (which reduce to 1/12), fith guy has to pull the trigger only 2/6 which gives him a measly 2/36 (reduces to 1/18).

That was the spinning case.

In case there is no spin after the first pull of the trigger:

first guy has 1/6 chance of shooting his brains out, which is the same as the last example. Second guy has 5/6 chance of having to shoot, and 1/5 chance of blowing his brains out if he has to fire, for a combined 1/6 (same as the first guy). Guy two has to fire 4/6 and has 1/4 chance of blowing of his head if he has to fire, giving a total of 1/6.

The numbers work through all the numbers.

While it is true that the last person is fucked if he has to shoot, there is also only 1/6 chance he has to shoot, so his odds are as good as the rest of them.

The problem with having to spin the gun in between is that you break the dependent variable that increase the chance of the gun going of on each non-bang but keep the one that lowers the properbility of having to fire.