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The Monty Hall Problem - where does our intuition fail us?

Marliyn vos Savant on September 9, 1990 wrote that the "correct" answer to this question was to switch doors, because switching doors gave you a higher probability of winning the car (2/3 instead of 1/3).

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? — Craig F. Whitaker Columbia, Maryland

The basics of this logic puzzle have been repeated more than once.

A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male? — Stephen I. Geller, Pasadena, California

Say that a woman and a man (who are unrelated) each has two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has two boys do not equal the chances that the man has two boys? My algebra teacher insists that the probability is greater that the man has two boys, but I think the chances may be the same. What do you think?

The question has even tripped up our very own Jeff Atwood. He posed this question:

Let's say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?

Jeff goes on to argue that it was a simple question, asked in simple language and brushes aside the objections of some that say that the question is incorrectly worded if you want the answer to be 2/3.

Is Marilyn vos Savant right, is the answer to these questions 2/3? (What assumptions do you have to make to come to that answer?)

Is the answer to these questions that you have a 50% chance (of getting the car, the other child being a girl, etc.)? (What assumptions do you have to make to come to that answer?)

Is the correct answer, "I don't know"? (What assumptions have to be clarified to be able to answer the question?)

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  • This is off topic, perhaps you want to ask on [maths.se]... – Sklivvz Jun 14 '12 at 06:27
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    Monty Hall is one of those "counterintuitive" things, which is why people have problems understanding it. Here is a video that explains it in a simple way. Plus, the MythBusters have tested it and gave it a Confirmed. –  Jun 14 '12 at 08:51
  • “tripped up our very own Jeff Atwood” – do you have references for that? The link you posted is related, but doesn’t corroborate this particular claim. As for the objections you link to, while cute, they have no basis in statistics. Jeff didn’t invent this problem, he just used it; its solution is well established in statistics. (Not that it matters, as this is entirely off topic.) – Konrad Rudolph Jun 14 '12 at 12:08
  • @KonradRudolph, it tripped him up because he claimed the answer is 2/3 without realizing that the question as worded is ambiguous, and you cannot come to that conclusion. –  Jun 14 '12 at 15:10
  • @user1873 The question is not worded ambiguously though. It’s counter-intuitive but unambiguous and completely straightforward once you understand conditional probabilities and the difference between p(A|B) and p(B|A). The blog you link to which claims otherwise is simply wrong. But I’ve in the meantime found a comment by Jeff saying that he initially got the wrong result (i.e. 1/2 instead of 2/3). – Konrad Rudolph Jun 14 '12 at 15:52
  • @KonradRudolph, the people who believe the answer is 2\3 cannot usually see the ambiguity (so I not surprised by your response). In all these questions, why\how you know certain pieces of information will determine the odds. Does Monty Hall always open a door? Does he always open a losing door? How did the vet determine at least one was a male puppy, did he lift one up, or both? How did you know the woman has at lest one boy, did you see one of them?Why did the woman with 2 children, tell yo one was a girl (100% the other is a boy since people don't talk like that). Ambiguity abounds. –  Jun 14 '12 at 20:25
  • @KonradRudolph, they most certainly do not collapse into the same result. Take for instance the question where you know an unrelated woman and man with two children and for the woman you, "know at least one is a boy." If we know this because we looked over the fence and saw one of her children, then she has the same odds as the man with the oldest boy (this is sometimes referred to as the Nebulous Neighbor) – user1873 Jun 15 '12 at 14:11
  • @user1873 I might have misunderstood the objection. If you look at all families, arbitrarily announce one child’s gender and then look at the probability of then having two children of said gender then the chance is indeed 1/2. But this is a fundamentally different question from saying “one of the children is a boy” which would constitute a different prior. But if you pick a random family, announce the gender of one child, and if that child is a boy look whether both are, then the probability is 1/3, and it doesn’t matter that you picked from a pool of random families. – Konrad Rudolph Jun 15 '12 at 14:15
  • I don't think this is an exact duplicate of the other Monty Hall question. For one, that question assumes that 2/3 is the correct answer, when it clearly is not. Perhaps I will roll this question into a proper answer to the "duplicate" question. – user1873 Jun 15 '12 at 14:15
  • "But if you pick a random family, announce the gender of one child, and if that child is a boy look whether both are, then the probability is 1/3" , but that is a different question. The.question only tells you that the result of randomly announcing the sex of one of the children was boy(50% chance the other is a boy too). It doesn't clearly state that "if you randomly chose a girl, move on to the next family". Like I said ambiguity. – user1873 Jun 15 '12 at 14:23

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