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[RedWarrior0]

I was doubting before I read that wikipedia article. It's basically this: You pick door 1. P(1) = 1/3; P(2 || 3) = 2/3; The result of 2 doesn't change that P(2 || 3) = 2/3 because OR is inclusive.

[Kagus]

You're saying the chance of picking 1 door in 100 is 1/2?

No, I'm saying picking 1 door in 100 is 1/100, and picking 1 door in 2 is 1/2.  That is the fundamental confusion regarding this problem.


You are given 100 choices, only one of them is correct, each one has an equal chance of being correct, and one of them must be correct.  Regardless of which door you choose, you will have a 1% chance of picking the right one.  This is the first stage of the problem.

Now we go through a lot of words regarding goats, sportscars, and a gameshow host.  98 choices are eliminated.

Now, how many doors are there?  2 doors.  One door has a car behind it.  None of the other 98 choices exist, they don't exist, and they never existed because this is a new problem.  You are given a choice between two doors.


Let's start off fresh.  I will give you two doors to pick from.  One of them has a prize, the other does not.  I guarantee you that one of the doors has a prize behind it, and that the other door does not.  Now, what is the chance for each door to be the right one?

If we'd just started with this, you would have said 50%.  But somehow, in a land far away, some chap herding 98 goats away from a TV studio manages to affect this probability.


Going in a completely different direction.  A host comes forward with 100 doors.  But before he lets you pick one, he eliminates 98 of them.  Now, what is the likelihood of picking the right door?


I really don't know how else to explain this.  It is not a two-part question, it is two questions.  The first question asks you which of the 100 doors is correct.  We never get an answer to this, and instead move on to the next question:  Which of the 2 doors is correct.

I've seen the Wikipedia page, I've seen the problem outlined in Discover, and I've heard it referenced or mentioned in a number of places on the web.  Not once, ever, have I seen a good explanation as to why the logic used in this problem is supposed to work.

[uber pye]

my PeTA:

People eating Tasty Animals

[Criptfeind]

Kagus: When the other doors are eliminated it has a 99/100 chance to bring it down to your door and the right one or a 1/100 chance to bring it down to the right one (yours) and a random one.

People eating Tasty Animals

I support peta.

People eating tasty animals is a great group.

[RedWarrior0]

I really don't know how else to explain this.  It is not a two-part question, it is two questions.

This is where you mess up. The opening of the doors is a smokescreen.

There are two possibilities. Either the door you picked originally is correct (1/3), or one of the other doors is correct (2/3). Imagine the host let you switch to ALL of the other doors. Would you switch?

Wikipedia's winning image:


PeTA of course, would not so as to save a goat from its horrible life as a game show prize.

[FuzzyZergling]

I think I have it:

The chances of you picking the wrong door the first time is more likely than you picking the correct door the first time.
Therefore, at the start of part two, you are more likely to have the incorrect door.
Therefore the door that you did not choose is more likely to be the correct door.

[RedWarrior0]

Congratulations, you just interpreted the image I just posted correctly. A winner is you!

[Eagleon]

  Psychology will march on without you. The scale will inevitably be changed, likely revised many times, dividing into groups and who knows what else. But you aught to start with some measure, and we do have methods of measure for single individuals of a species. Just like humans, scores will change with time and conditions.
 Or DJ's mockery, whichever you prefer.

Psychology is leaning towards useful intelligence being an artifact of sensory feedback and emotional learning, rather than a thing that can exist by itself. You can say, for instance, that a person that knows everything about calculus is smarter than a person that writes books, at any level, and be perfectly right in some models. Then you can turn around and select another that says the writer is exhibiting more advanced intelligence. The problem is, the complexity or uniqueness or whateverness of a behavior is very difficult to quantify objectively. We're better off studying the common origin of both behaviors. Trying to produce a measure for intelligence is meaningless when we still don't completely understand its origin, like trying to decide what color gravity is.

Situational analysis is far more useful for something with needs than accumulated facts (look at CYC, or any number of other expert systems, compared to just about any animal or human), and both require motive and sensation. We happen to excel at both, but you don't learn to do anything without rewards. No stimulus (internal or external) = no thought. If you're never hungry, you never eat, and if you never eat, or drink, or sleep, or feel pain, you're not alive or sapient in any sense of the word.

(sorry to derail the derailment o.o *steps aside*)

[Bauglir]

Kagus: They aren't separate problems, because the host's opening of doors is more likely than not to give you information about what is behind the other doors, which means you're not down to equal probabilities any more. There are 99 situations in which the host told you which door it was behind, and 1 situation in which he told you nothing about which door it was behind (because he only picked one door, not two, and he ALWAYS picks the correct door unless you got it right).

[dragnar]

I think I have it:

The chances of you picking the wrong door the first time is more likely than you picking the correct door the first time.
Therefore, at the start of part two, you are more likely to have the incorrect door.
Therefore the door that you did not choose is more likely to be the correct door.

Exactly. As I said, switching is essentially betting that you were wrong on your first guess, because ONLY if you were wrong originally(very likely) will the other door contain the prize.

[Eagle_eye]

I'm probably somewhere in the middle of the author and what he describes. Obviously animals aren't as intelligent as a human, but that doesn't change the fact that they can feel pain and suffer. Animal voting rights, no, not slaughtering millions for food, yes.

[G-Flex]

I think I have it:

The chances of you picking the wrong door the first time is more likely than you picking the correct door the first time.
Therefore, at the start of part two, you are more likely to have the incorrect door.
Therefore the door that you did not choose is more likely to be the correct door.

The reason you ought to switch is because the game eliminates one of the wrong doors for you; in the 2/3 chance that you picked the wrong door the first time, the host helpfully tells you which of the remaining doors is correct by eliminating the one that's not. The key thing here is that the host provides you with new information: You know that, in the case that the other two doors contain the correct door, he's picking the right one for you. If he didn't, then obviously being able to choose again would be meaningless.

[RedWarrior0]

I think I have it:

The chances of you picking the wrong door the first time is more likely than you picking the correct door the first time.
Therefore, at the start of part two, you are more likely to have the incorrect door.
Therefore the door that you did not choose is more likely to be the correct door.

The reason you ought to switch is because the game eliminates one of the wrong doors for you; in the 2/3 chance that you picked the wrong door the first time, the host helpfully tells you which of the remaining doors is correct by eliminating the one that's not. The key thing here is that the host provides you with new information: You know that, in the case that the other two doors contain the correct door, he's picking the right one for you. If he didn't, then obviously being able to choose again would be meaningless.

What? No, what the problem is is essentially you being asked whether you want two opportunities or one, but hidden behind a smokescreen. Since there is only one correct possibility, at least all but one of the two must be false, but the group as a whole has the same (2/3) probability. There is no elimenation, just a wonderfully crafted misdirection and a choice between two (or 999,999 or whatever) opportunities and one shot.

[ed boy]

You may find it easier to consider the following series of tables.  In each one, there are n doors, you pick one, the host opens doors other than yours until there are two left. If it is better to switch your choice, I will put W. If it is better to stick with your choice, I wll put T. The columns will be the door the prize is behind, and the row will be your choice.

Code: [Select]

n=2
+-+-+-+
| |1|2|
+-+-+-+
|1|T|W|
+-+-+-+
|2|W|T|
+-+-+-+

n=3
+-+-+-+-+
| |1|2|3|
+-+-+-+-+
|1|T|W|W|
+-+-+-+-+
|2|W|T|W|
+-+-+-+-+
|3|W|W|T|
+-+-+-+-+

n=4
+-+-+-+-+-+
| |1|2|3|4|
+-+-+-+-+-+
|1|T|W|W|W|
+-+-+-+-+-+
|2|W|T|W|W|
+-+-+-+-+-+
|3|W|W|T|W|
+-+-+-+-+-+
|4|W|W|W|T|
+-+-+-+-+-+

As can be seen, for each door the prize is behind, there is only one option out of n that will make you better off if you end up sticking. All the other options, you will get the prize if you stick. Sticking will have a winning chance of ¹/_n and switching will have a winning chance of ^n-1/_n

[Leafsnail]

If anyone want a sortof logic puzzle (although not nearly as hard):

I have a set of four cards.  On each card, there is a number on one side and a shape on the other.  I want to test the hypothesis

"If a card has a square on one side, it has an even number on the other side".

The sides I can see show 13, a square, 28 and a triangle.  Which 2 cards should I flip over in order to confirm or deny the hypothesis?