Answer: Here, you have better chances of survival if you chose to spin the barrel.
1) If bullets are placed in the adjacent chambers:
Let us do this step by step. The possible combinations for the bullets are:
1, 2
2, 3
3, 4
4, 5
5, 6
6, 1
Now if you ask not to spin the barrel:
The first shot went empty and thus the events (6, 1) and (1, 2) dont have bullets in them. Now only four possible shots are available.
Thus, P (death) = 1/4 = 0.25
P (Survival) = 1- 0.25 = 0.75
If you ask to spin the barrel:
Since you were not hit in the first shot, therefore
P (death) in the second shot = 2/6 = 0.33
P (Survival) in the second shot = 1 - 0.33 = 0.77
Clearly, you will have better chances of survival if you ask him not to spin the barrel.
2) If bullets are not placed in the adjacent slots:
If you dont spin it:
You were not hit in the first shot. Thus,
P (Death) in second shot = 2/5 = 0.4
P (Survival) in second shot = 1 - 0.4= 0.6
If you spin the barrel
P (death) in the second shot = 2/6 = 1/3 = 0.33
P (Survival) in second shot = 1 - 0.33 = 0.77
Here, you have better chances of survival if you chose to spin the barrel.