Here is the puzzle:

Mr. Black, Mr. Gray, and Mr. White are fighting in a duel. They each get a gun and take turns shooting at each other until only one person is left. Mr. Black, who hits his shot 1/3 of the time, gets to shoot first. Mr. Gray, who hits his shot 2/3 of the time, gets to shoot next, assuming he is still alive. Mr. White, who hits his shot all the time, shoots next, assuming he is also alive. The cycle repeats. If you are Mr. Black, where should you shoot first for the highest chance of survival?

The answer they give is

Mr. Black should shoot at the ground, guaranteeing he doesn’t hit either Gray or White. It will then be Gray’s turn to shoot and obviously Gray is going to kill White since White is going to kill Gray when it is his turn to shoot (because Gray has a higher likelihood of shooting White than Black does). If Gray kills white with his shot (2/3 chance), then it’ll be Black and Gray shooting each other until someone eventually dies. If Gray misses White then White kills Gray in his first shot, and Black has one shot then 33% chance of killing White, and then you’re dead if you miss.

I disagree with the answer though because

First of all this is kind of a dumb riddle because it’s a “gotcha”. They didn’t list the ground as a possible target so I guess you’re supposed to deduce it. The riddle should have just explicitly stated that each man has the option not to fire his gun at all. But let’s assume for a moment that aiming at the ground is just the logical equivalent of saying aim at Not Gray & Not White, which is fine.

The premise says that Mr. Black makes his shot 1/3 of the time. If Black’s target is the “ground” (aka no target), he only has a 1/3 probability of hitting the “ground” (aka nothing), and that leaves a 2/3 probability that Black will actually hit one of the two men instead (the only other things to hit, presumably).

That means he’ll have a

2/3[probability of not hitting ground]*1/2[probability of hitting one of the two other targets left to hit] = 1/3

probability of killing Gray if Black aims at the ground, the same probability that Black would have if he aimed at him directly. So their solution of aiming at the ground does not improve your odds of living if you are Black. The best thing to do is either aim at the ground or at Gray. Aiming at the ground has no benefit over aiming at Gray directly.

So what do you think? Am I wrong?