The Monty Hall problem

[Teohrn]

I guess a lot of you have already heard of it, and might even know the answer. It was asked and solved a long time ago. Still, I'd like the members of INTPf to solve it.

Using any source but your own mind isn't allowed; only independent thought is allowed. Some sort of explanation for your answer is preferred: simply answering yes or no isn't adequate.

Here's the question.

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

 

[pjoa09]

Yeah? Sheer probability?

 

[Teohrn]

Yeah? Sheer probability?

You can give whatever answer you like. Probability is basically what the problem is about. But if you or anyone else somehow manage to make it into a philosophical question, then be my guest. Everything is fine as long as it tackles the problem. Certain ways are better than others though, f.e. probability is more relevant than philosophy here. Whatever your answer may be, please reason for it, simply dropping some numbers is boring.

I guess I said a lot more than I had to, but I've been up for more than 2 days now, so I hope I'm excused.

 

[A22]

Doors: A B C
Only one door is the right door
You pick C

Probability that A or B are the right door (Pab Right):
2/3

That is equal to the probability that you have chose a wrong door.

The host knows which door is the right door, so he always opens a wrong door.

After he tells you that A is a wrong door:
Pab Right -> Only b left -> Pb Right 2/3

Pick B if possible, odds are it is the right door.

This video does a better job than me explaining it:
http://www.youtube.com/watch?v=mhlc7peGlGg

 

[Teohrn]

Unnecessary, but I'll just take a hopeful precautionary measure.

Spoiler

Basically - yes. When you first chose a door, the probability of you picking a goat was 2/3, then there is the revealing of one alternative door with a goat, here comes the problem for most, they think that the probability becomes different, as if the puzzle had always been two doors, one with a goat and one with a car, the probability as it was at the start still remains, however, as you have already taken a choice that was 2/3 likely to be wrong, by switching you are therefore 2/3 likely to pick the right door due to your original 2/3 probability to pick the wrong door.

Too bad this got solved that quickly, I would have thought it would last for longer. I've seen lots of people discussing this over great amounts of pages trying to find out the right answer, and some know the solution yet they don't agree with it, both probably understanding it as well as the other, although it could be argued that the former could understand the solution if given the solution.

1+ for INTPf I guess.

 

[Proletar]

The leverage lies in the fact that the hoast ALWAYS opens a goat-door - the hoast knows.


This inverts the probability. (C = prize)

If you choose A, he opens B.
If you choose B, he opens A.
If you choose C, he opens either A or B.

Which is 1/3 for the prize if you stay with your chosen one and 2/3 if you switch.

 

[Jennywocky]

Unnecessary, but I'll just take a hopeful precautionary measure.

Well, as far as things seeming to be solved quickly, chances are the mentality of the people on this forum already led them to trying to figure out this problem earlier in life, before the discussion started here. I ran across it a few years ago.

In forums, where no one's really tried to address the problem before, I'm sure the discussion would bounce around for awhile. The answer seems intuitive in one sense and counter-intuitive for another.

From my time with math at college, it's all in how the problem happens to be framed, in terms of statistical outcome -- it reminds me of the Shared Birthday in the Room problem in how there's a 50% chance of two people in a room with 21-25 people in it to share a birthday, and the probability shoots way up just by doubling that number. The "common sense" answer would try to say the odds are 1/365 * 1/365 or something, but that's actually the solution to a different problem and isn't the question being asked .

 

[7even]

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

Seems like a trick question..

Depends whether you want a goat or a car too. I like goats. Cars pollute the atmosphere anyway.

There's a probability of 33.33% of picking a car and a probability of 66.66% of picking a goat.

Once the host opens 1 of the 3 doors, which has a goat, you're left with an equally likely chance of either picking a goat or a car.

So from there it's up to chance, and switching your choice of doors will either get you the car or it won't. 50/50 - there's no logical reason to switch your choice of doors.

Although, perhaps there's something here about the goats, I'm going to presume that there must be a gap of 1 door between the goats; or else they'd somehow sense each others presence by animal instincts, and 'baa!' - which would allow the person choosing the door to know where the other goat is by using his senses (hearing).

So, in this scenario, put together with my presumption, it would be at your disadvantage to swap doors from door #1 to door #2.

UNLESS, goats remain calm when they are aware of each others presence in close proximity, stopping them from 'baa-ing' - which would mean it would be at your advantage to switch from door #1 to door #2.

But how the fuck would I know? I know absolutely nothing about goats.

EDIT: Shit, just watched the video explaining it... That is bloody counter-intuitive... The fuck am I doing talking about goats senses? (Was thinking about the extra senses animals possess...)