First, let me reitterate: I agree wholeheartedly that if you know a man has 3 children and you are told that at least 2 of the children are boys, the probability that the third child is a girl is 75%. Unfortunately, that's not how the problem was stated. Nor was the problem stated ambiguously. Quite simply, that's the problem that was intended to be stated, but unfortunately, a completely different problem was stated.
I think I came up with the best way to show, by analogy, how the statement of the 3 children problem is incorrect.
I have in a hat 3 cards:
1 is black on both sides
1 is red on both sides
1 is black on one side and red on the other side.
If I state "I looked at all three cards and decided to keep one of them. The card I kept has at least 1 side that's black" then we can deduce that it's either the BB card or the BR card. There's a 50% chance that the other side is black, and a 50% chance that the other side is red.
That's how you guys are interpreting the problem. Unfortunately, I disagree with that interpretation.
If I instead tell you to pull out a card and look at one side only, and you see a black side, then the probability is no longer 50/50 that the other side is red/black. We're now looking at one side out of 6; each of those sides is, of course, attached to another side. In 2 of those cases, we see a black side, the other side is black. In only 1 case out of the 6, we'll see black and the other side is red.
Thus, if we look at one side of 1 cards and see black, then there's a 2/3 chance that the other side is black, and only a 1/3 chance that the other side is red.
Now, suppose we know that there are millions of these cards out there; 1/3 of each type. And, I spot one on the ground and a black side is showing. There's a 2/3 chance that the other side is black.
This is the same situation as with the family - you're not simply told that the man has at least 2 boys. Instead, you see two boys with the man. At first, last night, I wasn't 100% certain of my solution. Right now, I'm 99.99% certain.
***
If you understand the above problem with the cards (It's a fairly well known intro statistics and probability problem), then allow me to make up a similar problem. You have 3-sided cards (shaped like brazil nuts)
On one of those cards, the sides are black, black, and black
On 3 of those cards, the sides are black, black, green
on 3 of those cards, the sides are black, green, green
and on 1 of those cards, the sides are green, green green.
You are told that I have a card and at least 2 of the sides are black. Conclusion: 3/4 chance that the other side is green. (Get it yet? BBG)
However, you are NOT told that at least 2 of the sides are black. Instead, you are told to select a card at random and look at two of the sides. You see black and black. What's the probability that the 3rd side is green, and what's the probability that the 3rd side is black. Assuming you understood the 2 sided card problem, I assume you can calculate that it's 50/50. Or, instead of pulling a card out at random, you see one laying on the street, next to a man. Unfortunately, being shaped like a brazil nut, you can't see the third side. But, you see two of the sides. And, those sides are BB. There's a 50% chance the other side is Boy and a 50% chance the other side is Girl.
***
And lastly, so that I can cross all my t's and dot all my i's, I bothered to look at the link about Marilyn Vos Savant<div class="FTQUOTE"><begin quote>
You meet a woman, and ask how many children she has, and she replies "two." You ask if she has any boys, and she replies "yes." After this brief conversation, you know that the woman has exactly two children, at least one of whom is a boy. When the question is interpreted this way, the probability that both of her children are boys is 1/3, as Marilyn has claimed.
You meet a woman and her son. You ask the woman how many children she has, and she replies "two." So now you know that this woman has exactly two children, at least one of whom is a boy. When the question is interpreted this way, the probability that both of her children are boys is 1/2, as Eldon has claimed.
Because the question does not state how we obtained the information, the question is ambiguous. </end quote></div>
Except, in this thread, it *IS* stated how we obtained the information.
Convinced yet?
p.s. If it makes you feel better, I'll bet that at least some of the original 50% people happened on the correct answer through incorrect logic.
edit:
Originally posted by: Nathelion
I just wanted to watch people mess up and not get it at all for weeks and weeks while I smugly smile into my beard and laugh and feel superior, and you just pop the answer out like that? Horrible manners:evil:
Still feeling superior?