bigjeff5 comments on Bayes' Theorem Illustrated (My Way) - Less Wrong

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Comment author: bigjeff5 22 December 2013 05:08:17PM *  -1 points [-]

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

I know it's not the be all end all, but it's generally reliable on these types of questions, and it gives P = 1/2, so I'm not the one disagreeing with the standard result here.

Do the math yourself, it's pretty clear.

Edit: Reading closer, I should say that both answers are right, and the probability can be either 1/2 or 1/3 depending on your assumptions. However, the problem as stated falls best to me in the 1/2 set of assumptions. You are told one child is a boy and given no other information, so the only probability left for the second child is a 50% chance for boy.

Comment author: EHeller 22 December 2013 05:16:21PM *  1 point [-]

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

Did you actually read it? It does not agree with you. Look under the heading "second question."

Do the math yourself, it's pretty clear.

I did the math in the post above, enumerating the possibilities for you to try to help you find your mistake.

Edit, in response to the edit:

I should say that both answers are right, and the probability can be either 1/2 or 1/3 depending on your assumptions.

Which is exactly analogous to what Jiro was saying about the Tuesday question. So we all agree now? Tuesday can raise your probability slightly above 50%, as was said all along.

However, the problem as stated falls best to me in the 1/2 set of assumptions. You are told one child is a boy and given no other information, so the only probability left for the second child is a 50% chance for boy.

And you are immediately making the exact same mistake again. You are told ONE child is a boy, you are NOT told the FIRST child is a boy. You do understand that these are different?

Comment author: bigjeff5 22 December 2013 05:17:40PM 0 points [-]

Re-read it.

Comment author: bigjeff5 22 December 2013 05:23:58PM *  -2 points [-]

The relevant quote from the Wiki:

The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy? One description of the problem states that we look into a window, see only one child and it is a boy. This sounds like the same assumption. However, this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a boy, then replacing). Let's call the statement "the sample is a boy" proposition "b". Now we have: P(BB|b) = P(b|BB) * P(BB) / P(b) = 1 * 1/4 / 1/2 = 1/2. The difference here is the P(b), which is just the probability of drawing a boy from all possible cases (i.e. without the "at least"), which is clearly 0.5. The Bayesian analysis generalizes easily to the case in which we relax the 50/50 population assumption. If we have no information about the populations then we assume a "flat prior", i.e. P(GG) = P(BB) = P(G.B) = 1/3. In this case the "at least" assumption produces the result P(BB|B) = 1/2, and the sampling assumption produces P(BB|b) = 2/3, a result also derivable from the Rule of Succession.

We have no general population information here. We have one man with at least one boy.

Comment author: EHeller 22 December 2013 05:39:35PM 1 point [-]

I'm not at all sure you understand that quote. Lets stick with the coin flips:

Do you understand why these two questions are different: I tell you- "I flipped two coins, at least one of them came out heads, what is the probability that I flipped two heads?" A:1/3 AND "I flipped two coins, you choose one at random and look at it, its heads.What is the probability I flipped two heads" A: 1/2

Comment author: bigjeff5 22 December 2013 05:48:50PM *  0 points [-]

Yes, and we are dealing with the second question here.

Is that not what I said before?

We don't have 1000 families with two children, from which we've selected all families that have at least one boy (which gives 1/3 probability). We have one family with two children. Then we are told one of the children is a boy, and given zero other information. The probability that the second is a boy is 1/2, so the probability that both are boys is 1/2.

The possible options for the "Boy born on Tuesday" are not Boy/Girl, Girl/Boy, Boy/Boy. That would be the case in the selection of 1000 families above.

The possible options are Boy (Tu) / Girl, Girl / Boy (Tu), Boy (Tu) / Boy, Boy / Boy (Tu).

There are two Boy/Boy combinations, not one. You don't have enough information to throw one of them out.

This is NOT a case of sampling.

Comment author: EHeller 22 December 2013 06:36:24PM *  2 points [-]

As long as you realize there is a difference between those two questions, fine. We can disagree about what assumptions the wording should lead us to, thats irrelevant to the actual statistics and can be an agree-to-disagree situation. Its just important to realize that what the question means/how you get the information is important.

We don't have 1000 families with two children, from which we've selected all families that have at least one boy (which gives 1/3 probability). We have one family with two children. Then we are told one of the children is a boy, and given zero other information.

If we have one family with two children, of which one is a boy, they are (by definition) a member of the set "all families that have at least one boy." So it matters how we got the information.

If we got that information by grabbing a kid at random and looking at it (so we have information about one specific child), that is sampling, and it leads to the 1/2 probability.

If we got that information by having someone check both kids, and tell us "at least one is a boy" we have different information (its information about the set of kids the parents have, not information about one specific kid).

This is NOT a case of sampling.

If it IS sampling (if I grab a kid at random and say "whats your Birthday?" and it happens to be Tuesday), then the probability is 1/2. (we have information about the specific kid's birthday).

If instead, I ask the parents to tell me the birthday of one of their children, and the parent says 'I have at least one boy born on Tuesday', then we get, instead, information about their set of kids, and the probability is the larger number.

Sampling is what leads to the answer you are supporting.

Comment author: bigjeff5 22 December 2013 06:54:39PM 0 points [-]

The answer I'm supporting is based on flat priors, not sampling. I'm saying there are two possible Boy/Boy combinations, not one, and therefore it takes up half the probability space, not 1/3.

Sampling to the "Boy on Tuesday" problem gives roughly 48% (as per the original article), not 50%.

We are simply told that the man has a boy who was born on tuesday. We aren't told how he chose that boy, whether he's older or younger, etc. Therefore we have four possibilites, like I outlined above.

Is my analysis that the possibilities are Boy (Tu) /Girl, Girl / Boy (Tu), Boy (Tu)/Boy, Boy/Boy (Tu) correct?

If so, is not the probability for some combination of Boy/Boy 1/2? If not, why not? I don't see it.

BTW, contrary to my previous posts, having the information about the boy born on Tuesday is critical because it allows us (and in fact requires us) to distinguish between the two boys.

That was in fact the point of the original article, which I now disagree with significantly less. In fact, I agree with the major premise that the tuesday information pushes the odds of Boy/Boy closer 50%, I just disagree that you can't reason that it pushes it to exactly 50%.

Comment author: EHeller 22 December 2013 09:25:30PM *  2 points [-]

Is my analysis that the possibilities are Boy (Tu) /Girl, Girl / Boy (Tu), Boy (Tu)/Boy, Boy/Boy (Tu) correct?

No. For any day of the week EXCEPT Tuesday, boy and girl are equivalent. For the case of both children born on Tuesday you have for girls: Boy(tu)/Girl(tu),Girl(tu)/Boy(tu), and for boys: boy(tu)/boy(tu).

That was in fact the point of the original article, which I now disagree with significantly less. In fact, I agree with the major premise that the tuesday information pushes the odds of Boy/Boy closer 50%, I just disagree that you can't reason that it pushes it to exactly 50%.

This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says "I was born on a Tuesday," that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called "sampling")

Do you agree that if, instead, the parents give you the information "one of my two kids is a boy born on a Tuesday", that this is a different sort of information, information about the set of their children, and not about a specific child?

Comment author: bigjeff5 22 December 2013 09:54:24PM *  0 points [-]

This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says "I was born on a Tuesday," that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called "sampling")

I agree with this.

Do you agree that if, instead, the parents give you the information "one of my two kids is a boy born on a Tuesday", that this is a different sort of information, information about the set of their children, and not about a specific child?

I agree with this if they said something along the lines of "One and only one of them was born on Tuesday". If not, I don't see how the Boy(tu)/Boy(tu) configuration has the same probability as the others, because it's twice as likely as the other two configurations that that is the configuration they are talking about when they say "One was born on Tuesday".

Here's my breakdown with 1000 families, to try to make it clear what I mean:

1000 Families with two children, 750 have boys.

Of the 750, 500 have one boy and one girl. Of these 500, 1/7, or roughly 71 have a boy born on Tuesday.

Of the 750, 250 have two boys. Of these 250, 2/7, or roughly 71 have a boy born on Tuesday.

71 = 71, so it's equally likely that there are two boys as there are a boy and a girl.

Having two boys doubles the probability that one boy was born on Tuesday compared to having just one boy.

And I don't think I'm confused about the sampling, because I didn't use the sampling reasoning to get my result*, but I'm not super confident about that so if I am just keep giving me numbers and hopefully it will click.

*I mean in the previous post, not specifically this post.

Comment author: EHeller 22 December 2013 10:18:28PM *  2 points [-]

Of these 250, 2/7, or roughly 71 have a boy born on Tuesday.

This is wrong. With two boys each with a probability of 1/7 to be born on Tuesday, the probability of at least one on a Tuesday isn't 2/7, its 1-(6/7)^2

Comment author: bigjeff5 22 December 2013 06:11:39PM 0 points [-]

For the record, I'm sure this is frustrating as all getout for you, but this whole argument has really clarified things for me, even though I still think I'm right about which question we are answering.

Many of my arguments in previous posts are wrong (or at least incomplete and a bit naive), and it didn't click until the last post or two.

Like I said, I still think I'm right, but not because my prior analysis was any good. The 1/3 case was a major hole in my reasoning. I'm happily waiting to see if you're going to destroy my latest analysis, but I think it is pretty solid.

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