Probability Puzzle - I disagree with their solution

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?

I think you’re reading too literally. Presumably aiming at the ground is a 100% likely shot no matter who pulls the trigger.

Of course, the idea that someone hits their target a precise percentage of the time is ridiculous too, but it’s a lateral thinking puzzle. You’re supposed to think outside the box.

That might be possible, if they hadn’t closed the goddam lid. The puzzle statement, if drewtwo99 has quoted it verbatim, says that the participants take turns shooting “at each other.” The ground is not on the list of acceptable targets. If they had left out the words “at each other,” then the ground would have been an option.

I think you’re correct that the puzzle is under-specified as a pure logic puzzle. However your objections are wrong in any common-sense interpretation.

A shooter accurate enough to hit his target 1/3 of the time is certainly accurate enough to miss reliably if he aims well away from both targets.

This puzzle setup is sometimes called “truel.”

If we are going to go with “common sense” answers, then the best answer to the question of how maximize your odds of surviving the duel is to just refuse to ever take a shot. This guarantees no one else ever gets a turn.

But that’s kind of a trivial, lame answer. So if shooting at the ground is allowed (again, I’m reading this more as shooting at Not Gray and Not Black which does seem like a valid logic option), then you have to apply the rules given, and the rules CLEARLY state that he only makes 1/3 of his intended shots.

I agree it’s a little sneaky, but the phrasing of the question “where should you shoot” raised my suspicions, and I assumed the answer was going to be anywhere but at a participant without bothering to work it logically through.

Incorrect. There is a whole universe of things to hit besides the other two men, even if you could somehow manage to miss the ground. I think we can assume that a shot intended to miss both other other men has a 100% chance of succeeding.

I said that I assumed targeting the ground was the logical equivalent of targeting Not Gray and Not Black. Why can we assume something that directly contradicts information given in the riddle (that black only hits his target 1/3 of the time). If a shot is intended to miss (aka targetting Not Gray and Not White), then there is a 2/3 probability that it will not miss, thus either hitting Gray or White.

This is from the solution that drewtwo99 disagrees with.

This step is too quick. If Mr. Gray passes, then, if Mr. White shoots Mr. Gray, he will have a 1/3 chance of a loss. But if Mr. White passes, we are back in the initial position; and if everyone made their best moves, we have a loop. I don’t know what we do with a loop.

Or all three participants could keep shooting at the ground indefinitely, so that everyone has a 100% chance at survival. I agree that the problem as stated is underspecified.

EDIT: TATG beat me to it.

Search Google Groups for “truel” to see that this puzzle, and objections to it, have been well beaten.

To this objection my response is:
Would you rather have a 1/3 chance (or so) of having a long fruitful life, or be doomed to spend your entire threescore and ten doing nothing but firing a gun at the ground. :dubious:

I’m criticising the (lack of) specification in the original post, not the idea in general. I’m sure there’s a way that you could make it rigourously clear what the actual rules are, but that would probably spoil the “gotcha” quality of the puzzle.

And Mr Grey’s best move is also to shoot the ground, because then Mr White’s best move would also be to shoot the ground because that resets the loop.

Once again proving that “the only winning move is not to play”…

I think White should shoot at Black.

If he hits, then he just has to win against Grey, which he would probably have to do anyway since he is such a bad shot.

If he misses, then its the same as having not aimed at anyone.

So I dont get why he would waste a shot at the best opponent.

Methinks you have White and Black mixed. The question is where should Black, the poorest shot, aim his shot?

I agree, though, that Black should aim for White (assuming that’s what you meant).

Black definitely wants White out of the picture. While White, if he gets a shot, should aim for Grey (who presents a bigger threat to him than Black does), if it ever comes to White having a shot after Grey is dead then Black is done for. So Isilder’s logic here makes sense – Black should aim for White.

The answer given in the OP seems to suggest that aiming for White, Black might hit Grey (which means death for Black) – there’s certainly no other reason for Black to waste a shot. But come on, it’s rather ridiculous to suppose that an errant shot aimed at White is somehow going to kill Grey. That’s just too precious.


Any reasonable reading of “makes his shot” refers to a specific target small enough to miss. To suggest that one might miss the ground when aiming for it is ludicrous.

Even if he did somehow miss the ground, that doesn’t mean he must hit one of the other men. The shot might hit a tree, or go through the air without hitting anything (maybe landing in a lake), or any number of other things that don’t require it to hit one of the other shooters. That was a HUGE assumption to make, with no basis for it that I can see.

I think it’s a huge assumption to make that he only hits 1/3 of targets that are “small enough to miss” considering we aren’t given ANY information about the duel. Maybe these men are just a meer 12 inches apart from one another. Who knows why they miss as often as they do?

And as I said SEVERAL times, the point of “at the ground” was that he was not aiming at the GROUND, but rather aiming at no one (logically speaking not Gray and not Black). You aren’t following the rules of logic if you start making up your own rules for the riddle.

If I aim at “Not Mr. Gray” then I have a 2/3 chance of hitting him, if my odds of hitting my target are 1/3, by a pure logical reading.

Similarly, if I aim at “not Mr. Gray & not Mr. Black” (logical equivalent of aiming at the ground), then I have a 2/3 chance of hitting either Gray or Black.

I don’t see what’s so hard to follow in this logic. Point out where the logic is wrong. Don’t just go making things up like assuming what a guy can or can’t hit based on silly assumptions about the size of targets.

Oh come now. If you don’t see that assuming that a shot aimed at the ground/nothing/whatever that misses said ground/nothing/whatever MUST hit one of the other shooters is patently ridiculous, I don’t know if there’s any hope. The puzzle as stated certainly didn’t specify that. It’s not out of line to assume normal Earth conditions for scenarios like this. To assume that bullets aimed at nothing must magically hit some target, however…

You guys are ignoring the order of the shots. If Black succeeds in killing white, gray gets the next shot, 2/3ds chance.

If gray is the one to kill white, then Black gets to go next.