Call Amos is A, Butch is B and Cody is 3

C_1:  - If A is the one to shoot first and shoot C, he will be killed by B

     => SO A will shoot B first

     Then it's time for C to shoot A

     => The percentage of A alive is 50% and C is always alive (out of shot)

 C_2: - If B is the one to shoot first and shoot C, he will be killed by A

     => SO B will shoot A first

     Then it's time for C to shoot B

     => The percentage of B alive is 50% and C is always alive (out of shot)

 - If C is the one to shoot first 

     + He shoots A:

          C₃. If shoot to target, B will kill C => B is alive

          C₄. If shoot miss, B will kill A or A will kill B (reason above) and then C and A (or B) are alive (out of shot)

  + He shoots B:

          C₅. If shoot to target, A will kill C

          C₆. If shoot miss, B will kill A or A will kill B (reason above) and then C and A (or B) are alive (out of shot)

SO WE HAVE:

          ALIVE        50%        DEAD

C₁:         C               A               B

C₂:         C               B               A

C_3:           B                              A , C

C_4:       C, A (or B)                B (or A)

C₅:         A                               B, C             

C₆:       C, A (or B)                B (or A)         

SO C HAS THE BIGGEST CHANGE TO LIVE (\(\dfrac{4}{6}=66.66\%\) )

     A (or B)' s probability to live;\(\dfrac{1-\dfrac{4}{6}}{2}=\dfrac{1}{6}=16.66\%\)

Call Amos is A, Butch is B and Cody is 3

C1:  - If A is the one to shoot first and shoot C, he will be killed by B

     => SO A will shoot B first

     Then it's time for C to shoot A

     => The percentage of A alive is 50% and C is always alive (out of shot)

 C2: - If B is the one to shoot first and shoot C, he will be killed by A

     => SO B will shoot A first

     Then it's time for C to shoot B

     => The percentage of B alive is 50% and C is always alive (out of shot)

 - If C is the one to shoot first 

     + He shoots A:

          C3. If shoot to target, B will kill C => B is alive

          C4. If shoot miss, B will kill A or A will kill B (reason above) and then C and A (or B) are alive (out of shot)

  + He shoots B:

          C5. If shoot to target, A will kill C

          C6. If shoot miss, B will kill A or A will kill B (reason above) and then C and A (or B) are alive (out of shot)

SO WE HAVE:

          ALIVE        50%        DEAD

C1:         C               A               B

C2:         C               B               A

C3:           B                              A , C

C4:       C, A (or B)                B (or A)

C5:         A                               B, C             

C6:       C, A (or B)                B (or A)         

SO C HAS THE BIGGEST CHANGE TO LIVE (46=66.66%

 )

     A (or B)' s probability to live;1−462=16=16.66%

AUTO ANSWER (AFTER SEVERAL DAYS): I don't khow why the moderator selected this answer. The answer is wrong.

If Amos and Butch shoot first, they will shoot each other (always on target). So Cody has 50% chance to win, while others has \(\dfrac{50\%}{2}=25\%\) chance to win because their probability at shooting is all 100% target.

If Cody shoots first, he'll shoot miss (because if he shoots on target, the other person will defeat him), let the others shoot each other. Then he'll have 50% chance to win, while others has only 25% chance to win.

So the probability that Cody is alive is 50% and the two others are 25%/ person.

It is correct, man