Sorry, correction odds of both being head is 1:2, had it reversed when typing.

Great thread, BTW semck. Provoked a lot of though and discussion. I'll donate a +1 to the thread ;-)

By the way, liked the use of bit comparison in the C++ version, but you could have really gotten tricky and bit shifted tallying up the positive bits to either 0,1, or 2 and defeated my argument about position in full because you still would have statistically twice as many with one positive bit than with two positive bits.

Yes, my coffee finally kicked in and I'm getting my second wind. Got a client who is running behind from another dev team, so they pulled me in as the "go to" guy who always gets things done. Time for a long weekend coding session.

As several have pointed out, total possible scenarios with two children are BB, BG, GB, and GG. We are working with only the subset BB, BG, GB, and of those three, only BB meets the criteria of a second boy child. Thus one in three.

Oh wait.

Yeah. It would be different if he already had a boy and you were asking what the chances of another boy being born, which would be 50/50.

Semck, you write "But spyman, we're talking about his second child. We already know he has one boy." Again, ambiguous and confusing. It should read "other child." lol

Well, it depends how you're enumerating, Mujus. "Younger" would be the unambiguous term for the second-born. In context and with the antecedent clarification, I'll stand by this one.

Boy, Boy
Boy, Girl
Girl, Boy

^ This is of how I was thinking in my first answer, Semck83. Getting rid of the notion of position, but assuming that at least one child is a boy. I saw four boys out of a total of six children thus 4/6 = 2/3. Which is what I was trying to explain for, but my head was spinning too much to put into words as the 50/50 option kept flashing before me too.

typo... explain *before (ie earlier)

If we look at the two columns we can see that in two out of the three spaces there are two boys. Hence 2 out of 3.

oops sorry, I mean two out of the three rows.

.. and of course everything I wrote makes no sense. But that I was thinking.

I am still confused. If I phrase the question like this: Mr Jones has a two children. One of them is a boy. What is the probability that the second child is a boy. Then is the answer 2/3 as per my diagram?

No, you have to treat each row separately, spyman. Your diagram is counting the boy we already know about three times.

Does my phrasing of the question change the result though Semck83 or are you saying it is still 1/3?

Mr Jones has two children
One of the children is a boy (it could be the younger or it could be the older)
Given what we know about Mr Jones... what is the probability that the second (ie younger) child is a boy?

Here are the options.
1. Mr Jones has two boys - the second child is a boy
2. My Jones has - an older girl and a younger boy - the second child is a boy
3. My Jones has a older boy and a younger girl - the second child is girl

There are no other possibilities. Given what we know about Mr Jones there is a 2/3 probability that that second child is a boy.

I looked this up and it is clear that the answer given to this question is 1/3 as you have argued, but I can't see the flaw in my reasoning. :-(

The best wording would have been "what are the odds that both children are boys" as that clarifies it is combination theory. And, spyman, semck correctly points out that the known boy is in each of the three combinations so you subtract three from your boy count giving you 1 boy for 2 girls or 33% chance.

If it had, conversely, been worded that "you have met one of his children, a boy. What are the odds of the other child being a boy?", then you reduce it to a known entity (positional) and can say 50/50 because the positions are in the order in which you meet them making it BB and BG as the only two options.

If you say "second" or "younger", then you are taking the first event out of the problem, and its irrelevant whether its a head, tail, girl or boy. The probability that the second is a boy is now 50%

Gotcha Draug... when you put it like that "what are the odds that both children are boys" - yes clearly 1/3. Alas that was not how interpreted the question. And given the question posed:

Suppose you know ONLY the following two facts about Mr. Jones and his family:

a) He has two children.
b) At least one of them is a boy.

^How can that be interpreted as "what is the probability of both children being boys". I don't follow that. Maybe the penny will drop later. I must be confused.

Read on. It goes on to ask, what is the probability his second child is a boy which was then changed to "other child". Both would by implication be asking what the probability was both were boys.

Ah! Mujus was indeed right that I was being confusing. By "second child," think "other child." So in your option 2, the "second child" is in fact a girl. The first child is the one we already know about -- the boy. (We weren't told if he was younger or older).

This is what I had in mind re some fathers being more likely to produce sons and some being more likely to produce daughters: http://www.ncl.ac.uk/press.office/press.release/item/boy-or-girl-it-s-in-the-father-s-genes#.UXtYJcpc3fU

Sorry, didn't refresh after opening this to see your question twenty minutes ago.

The use of the term second or other implies knowledge of the first (or not other) where as what is the probability both are boys has no implication that you can recognize one of the bnoys and that you are strictly looking at the remaining one.

Another way to think of it is I throw 2 dice. I knwo one is a 6 (nut not which one), what is the odds both are a 6? 10:1 against. What are the odds the second one or other one is a six? 5:1 against. You have knowledge to the specific dice being a six. But the combinations having a six are 11 out of 36 and 1 of those 11 have double 6.

You can also pose the question there are x children born to y parents in z different countries over d many days ... Ignoring ALL that, what is the probability that the next child born is a boy? ... 50%

But Draug, it is still the same if we change it to older rather than younger:

1. Mr Jones has two boys - the older child is a boy
2. My Jones has - an older girl and a younger boy - the older child is a girl
3. Mr Jones has a older boy and a younger girl - the older child is boy

Or

1. Mr Jones has two boys - the younger child is a boy
2. My Jones has - an older girl and a younger boy - the younger child is a boy
3. Mr Jones has a older boy and a younger girl - the older younger child is girl

However you define "other" the result is the same - 2/3

The key is knowledge of that order. Maybe it is order in which you met the kids. When you can identify a specific child as being a specific sex, then the odds become even. whne you cannot idemntify the child, only that one of them is a boy, then the odds of two boys are 2:1 against. When you don't know any of the children. the odds are 3:1 against.

Sorry everyone I am extraordinarily thick sometimes when I get an idea in my head. Maybe I will grasp this later on, when I look at it with fresh eyes.