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[kaijyuu]

I feel educated

[Lectorog]

You don't have to do that in this case, because picking again at random loses information that was encoded in the actions of the host.

But most people will. It is indeed a red herring, but in most cases it's true.

[Fenrir]

You don't have to do that in this case, because picking again at random loses information that was encoded in the actions of the host.

But most people will. It is indeed a red herring, but in most cases it's true.

wat

[Reelya]

@lectorog: we're discussing a specific logic puzzle, what's true in other puzzles is of no concern. Saying 50/50 is true in other situations is like objecting to a maths problem solution "x=5" because "i did another maths problem and x=0 was the result".

In this puzzle, swapping doors gives you a 2/3 chance of winning. It doesn't matter that "most people" will make the wrong decision, we're discussing what is the best decision to make.

===

Now, it would be 50/50 in a different situation like this: The host says "I'm feeling generous, so I'll reveal one of the goats before you even pick doors". In this different scenario, after he reveals a goat, it is indeed 50/50 whether you pick the right door in this case, but you have significantly less chance of winning than in the original scenario, so he wasn't actually being "generous" by eliminating one of your choices first.

[Lectorog]

I think I'm trying to apply probability to decision making rather than actual chances of winning. Either way, it seems to only be making sense in my head.
Trying something else:
If a given individual has a 1/2 probability of staying with a 1/3 chance of winning, and a 1/2 of going to 2/3, what does that make their overall probability? Does this represent the uneducated decision?

[rhesusmacabre]

They would have a (1/2*1/3)+(1/2*2/3)=1/2 chance of winning.

[Reelya]

Well, if the individual decides to just pick at random out of the remaining 2 doors, then they do in fact only have a 50/50 chance of getting it right, but they're missing out on maximizing their chances.

The real basis of the problem is that your only independent random decision was out of 3 possibilities, not 2. The second choice "switch or not switch" isn't a true independent trial.

That's because if you picked 2/3 goats, then after the host eliminates the other goat, you have a 100% chance of getting the car by switching. In the 1/3 times you did pick the car first, then switching is the wrong thing to do. What you get from switching is pre-determined by which of the 3 doors you picked at first, so it isn't random any more.

Another way to see that this must be true, is that you only had 1/3 chance of picking the car in the first place. So sticking with your door gives you 1/3 chance of winning, and 2/3 chance of losing. Switching doors is by definition the "mirror image" of these chances: 2/3 win, 1/3 lose.

[Montague]

I don't understand the problem. You pick another door and it will either have a car or a goat, either way win-win, free goat or free car.

[kaijyuu]

Well it's assumed you'd want the car more, even if just for resale value.

[Darvi]

Montague has it right. But for all the people who don't like goatmilk, -meat, or -feces, the car would be a better option.

Also I don't know what the fuss is about. When you switch, your new choice is ALWAYS whatever you didn't pick first - unless you're being incredibly stupid and pick the door that has been shown to have a goat behind it.
So because your choice will be always different, and because there was a 2/3 chance of getting a goat to begin with, you effectively have the same 2/3 chance to get the car.

[kaijyuu]

People get confused and don't realize the original door opening gave them information about the remaining two. It *would* be 50/50 if you didn't know which door was originally picked.

[Leafsnail]

What is happening becomes more obvious if you change the puzzle just a little.

Scenario: You're on a tacky game-show, you've survived the Wheel Of Death, beaten all the other contestants in the elimination rounds, and now you're up to the final challenge of the night.

The host speaks: "Before you are doors number 1, 2 and 3, 4, 5,[...] 99 and 100 - behind one of the doors is a car, the other 99 have goats. Which door do you want?"

You pick one of the doors, then the host opens 98 other doors to reveal 98 goats.

He speaks again: "Your door could have the car, or does the remaining door have the car? You have one chance swap doors right now before we reveal where the car is, it's 50/50."

What do you do? Do you stick with your door or swap doors? Does it even matter?

[Neonivek]

Scenario: You're on a tacky game-show, you've survived the Wheel Of Death, beaten all the other contestants in the elimination rounds, and now you're up to the final challenge of the night.

The host speaks: "Before you are doors number 1, 2 and 3, behind one of the doors is a car, the other two have goats. Which door do you want?"

You pick one of the doors, then the host opens another door to reveal a goat.

He speaks again: "Your door could have the car, or does the remaining door have the car? You have one chance swap doors right now before we reveal where the car is, it's 50/50."

What do you do? Do you stick with your door or swap doors? Does it even matter?

This is one of those things that makes little sense but are true.

If you stick with the door you are picking there is a 1 in 3 chance of success (or in essence 1 in 3)

If you do the deal there is a 2/3rd chance of success

[Pnx]

Ok, here's a slightly similar logic puzzle I enjoyed figuring out.

You're teleported to a magic realm where there are three gods in front of you, one of them always lies, one always tells the truth, and one randomly alternates between truth and lies. If you can figure out which is which then they will grant you any wish you desire, there's just a few complications here.

1. You may only ask yes or no questions.
2. They answer in a language you can't understand, the language has yes or no, but you don't know which is which.
3. You may only ask three questions, the first may be directed at all of them, the second and third may only be directed at one of them.

With these rules there's a way to work out which is which 100% of the time, what is it?

[Darvi]

I only get 50% :V