The Good, Bad, and Ugly Shootout | Solution

Did you guess that Ugly’s best bet is to shoot in the air? If not, then here’s how we arrive at this answer.

The main idea here is that Ugly has options. What if he leaves Good and Bad alive and lets them take each other down? Will his odds of survival be better? Or should he try to take someone out right away? This is what Ugly is calculating in the split second he has. And his life depends on it!

Get one obvious bad choice out of the way. We can simplify Ugly’s choice down to two things right away. This is because if he shoots at Bad and hits him, then Good will shoot next and Good will kill him. He’s not going to take that chance.

Consider firing in the air. Let’s really think first about the idea that Ugly can fire into the air. Now, it’s worth pointing out that technically, everyone could decide to fire into the air, which would mean everyone would survive. But that’s just silly. They wouldn’t be having a gun fight if they all wanted to walk away alive.

So, Ugly shoots in the air, deciding to let the stronger two duel it out.

Now, Bad shoots next. Bad shoots at Good, with a ⅔ chance of hitting him. If Bad hits Good, then Ugly shoots next, with a ⅓ chance of hitting Bad. If Ugly misses, then the two will duel.

If Bad misses Good, then Good will shoot Bad next, which means Ugly has a ⅓ chance to hit Good before he dies. At least he gets a ⅓ shot at it, and on top of that, he has the ⅔ chance that Good will be taken down.

Consider if Ugly shoots at Good. Ugly sees that he might be best to shoot in the air, but first, he wants to think through the possibility of taking the strongest guy down first. Now, he has a ⅓ chance of hitting Good.

If he kills Good, then it’s only Bad who’s left, but Bad will get the first shot in their duel (with a ⅔ chance of hitting Ugly).

It’s not much better if Ugly misses Good. Bad will shoot at Good next, but there’s a ⅓ chance he’ll miss, which means Good shoots next and Bad is dead. Then Ugly has one shot at Good (with only ⅓ odds), and if he misses, he’s dead. Bad might hit Good, which means Ugly gets to go first in their duel, but all in all, the other possibility that Good might be left to duel with Ugly makes this option overall less desirable.

And there you have it: logical proof that sometimes it’s best to let the toughest fight it out.

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