Flip two coins 1000 times, then count how many of those trials have at least one head (~750). Count how many of those trials have two heads (~250).

Flip two coins 1000 times, then count how many of those trials have the first flip be a head (~500). Count how many of those trials have two heads (~250).

By the way, these sorts of puzzles should really be expressed as a question-and-answer dialogue. Simply volunteering information leaves it ambiguous as to what you've actually learned ("would this person have equally likely said 'one of my children is a girl' if they had both a boy and girl?").

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?

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?

No, it's the exact same question, only the labels are different.

No, it isn't. You should consider that you are disagreeing with a pretty standard stats question, so odds are high you are wrong. With that in mind, you should reread what people are telling you here.

Now, consider "I flip two coins" the possible outcomes are hh,ht,th,tt

I hope we can agree on that much.

Now, I give you more information and I say "one of the coins is heads," so we Bayesian update by crossing out any scenario where one coin isn't heads. There is only 1 (tt)

hh,ht,th

So it should be pretty clear the probability I flipped two heads is 1/3.

Now, your scenario, flipped two coins (hh,ht,th,tt), and I give you the information "the first coin is heads," so we cross out everything where the first coin is tails, leaving (hh,ht). Now the probability you flipped two heads is 1/2.

I don't know how to make this any more simple.

"The first coin comes up heads" (in this version) is not the same thing as "one of the coins comes up heads" (as in the original version). This version is 50%, the other is not.

No one is suggesting one flip informs the other, rather that when you say "one coin came up heads" you are giving some information about both coins.

I have two coins. I flip the first one, and it comes up heads. Now I flip the second coin. What are the odds it will come up heads?

This is 1/2, because there are two scenarios, hh, ht. But its different information then the other question.

If you say "one coin is heads," you have hh,ht,th, because it could be that the first flip was tails/the second heads (a possibility you have excluded in the above).

I have flipped two coins. One of the coins came up heads. What is the probability that both are heads?

1/3 (you either got hh, heads/tails,or tails/heads). You didn't tell me THE FIRST came up heads. Thats where you are going wrong. At least one is heads is different information then a specific coin is heads.

This is a pretty well known stats problem, a variant of Gardern's boy/girl paradox. You'll probably find it an intro book, and Jiro is correct. You are still overcounting. Boy-boy is a different case then boy-girl (well, depending on what the data collection process is).

If you have two boys (probability 1/4), then the probability at least one is born on Tuesday (1-(6/7)^2). ( 6/7^2 being the probability neither is born on Tuesday). The probability of a boy-girl family is (2*1/4) then (1/7) (the 1/7 for the boy hitting on Tuesday).

This sounds like you are trying to divide "two boys born on Tuesday" into "two boys born on Tuesday and the person is talking about the first boy" and "two boys born on Tuesday and the person is talking about the second boy".

That doesn't work because you are now no longer dealing with cases of equal probability. "Boy 1 Monday/Boy 2 Tuesday", "Boy 1 Tuesday/Boy 2 Tuesday", and "Boy 1 Tuesday/Boy 1 Monday" all have equal probability. If you're creating separate cases depending on which of the boys is being referred to, the first and third of those don't divide into separate cases but the second one does divide into separate cases, each with half the probability of the first and third.

doesn't add any information. How could it?

As I pointed out above, whether it adds information (and whether the analysis is correct) depends on exactly what you mean by "one is a boy born on Tuesday". If you picked "boy" and "Tuesday" at random first, and then noticed that one child met that description, that rules out cases where no child happened to meet the description. If you picked a child first and then noticed he was a boy born on a Tuesday, but if it was a girl born on a Monday you would have said "one is a girl born on a Monday", you are correct that no information is provided.

What's the difference between

Boy1Tu/Boy2Tuesday

and

Boy1Tuesday/Boy2Tu

?

Here is the breakdown for the Boy1Tu/Boy2Any option:

Boy1Tu/Boy2Tuesda

Then the BAny/Boy1Tu option:

Boy2Tuesday/Boy1Tu

You're double-counting the case where both boys are born on Tuesday, just like they said.

A key insight that should have triggered their intuition that their method was wrong was that they state that if you can find a trait rarer than being born on Tuesday, like say being born on the 27th of October, then you'll approach 50% probability.

If you find a trait rarer than being born on Tuesday, the double-counting is a smaller percentage of the scenarios, so being closer to 50% is expected.