The problem is in this specific example, the knowledge of one being male (out of a possible two) leads us to jump to positional knowledge in our mind even though the question doesn't indicate that we have that knowledge. Wording with names and details do that to us. Wording with questions like "a random sample of two kids have at least one boy in the bunch, what are the odds both are boys" doesn't provide that level of detail and allows us to think in combinations.

I have just tested this again in Excel:

Two columns - (A) First and (B) Second child
If I randomly generate the the sex of the children for a hundred cases
=CHOOSE(RANDBETWEEN(1,2),"Boy","Girl")
We know that at least one child must be a boy thus we will exclude all rows were both children are girls:
=OR(A2="Boy",B2="Boy")
And I filter on the rows where the at least one child is a boy, I get approx 76 rows.

If I now count the number of boys in each column and compare then with the number of girls, the boys outnumber the girls 2:1.

Sex First Second
Boy 48 53
Girl 28 23

This experiment matches the perimeters of the question.
1. Mr Jones has two children
2. At least one is a boy
3. What is the probability that the other child is a boy.

Thus I conclude I am either right, or I have misunderstood the question. But given the question as I have posted above - 2/3 must be right, and can be demonstrated empirically.

What is wrong with this?

spelling error - matches the *parameters of the question

For every column that has a girl, the other column has a boy after the filter. This is the known boy. so you have to subtract the number of girls from the number of boys then divide by two to get the number of all boy pairs. You are counting individual children when you should be counting pairs.

So you had 51 mixed pairs (28+23). Add the boys together (101), subtract the pairs that had girls (101-51=50) and divide by 2 because there are two boys in the BB pair (50/2=25). You had 51 mixed and 25 BB pairs or 2:1 approximately.

@spyman: I assume you had 100 rows per filter? And by filtering out the GG combos you are left with 76. So far so good. That is the total number of outcomes given the assumption that they aren't noth girls. Now from that set - whih again is the total number of outcomes do an
And(A2="Boy",B2="Boy")
To get the number of favorable outcomes, remember the definition of Probability = fab/possible. You should get about 25. Which is about 3/4. Since youre doing this empirically, the larger number of rows, the closer you'll get to 25% favorable out of 75% total, then divide and you'll get 33%

That's trying to explain your excel results. But it's really easier when only dealing with 2 possible out comes of 2 events to just write them all out. These are the only possibilities:

GG
GB
BG
BB

the original assumption is that at least one is a boy, leaving you with

GB
BG
BB

three possible outcomes. Now of those, how many are favorable? That is, given one boy, the other is also a boy, in other words two boys. Only one of the three matches that criterion, therefore fav = 1 and possible = 3, so fav/pos = 1/3

Does that clear it up?

Per filter = pre filter

1/3

Actually if do accept the wording "other" meaning in addition to the known child then yes you are right we are talking about both children being boys. That would be 1 in 3.

In my example above I meant to say "What is the probability that the younger child (of the the two children)" being a boy.

Thus for example if we meet Mr Jones and see that he has one boy child. But we don't know if the child is the younger child or the older child (we don't know the ages of the children). If we are then asked to guess, based on what we know, whether Mr Jone's younger child is a boy... that is a different question. No? With a different answer. Or does it not make a difference?

Yes, spyman, that's another answer. That's 1/2.

Actually, I misread the question. I think 1/2 is wrong for the one you asked. Thinking.

Here is the question revised:

We we are walking down the street and we meet Mr Jones and he introduces us to his son. Later we are told that Mr Jones has two children. What is the probability that Mr Jone's younger son is a boy?

typo.... Mr Jones's

The chance that his younger son is a boy is 100%. ;-)

But yes, at that point, I would say the probability that his younger *child* is a boy is 2/3. We have

GB
BG
BB

In two of them, the younger is a boy.

sorry... that is crazy... not younger son.. younger child.

Obviously young son is 100% a boy.

Now want something completely disorienting?

Say you're walking down the street and you meet Mr. Jones, who you know has two kids, and he has a kid with him, and it's a boy.

What are the odds that the other child is a boy?

50%.

But I am not asking about the *other* child.

We we are walking down the street and we meet Mr Jones and he introduces us to his son. Later we are told that Mr Jones has two children. What is the probability that Mr Jone's younger *child* is a boy?

^ for some reason that is how I interpreted your original question.

Just to be clear - we don't know if the child we just met is the younger or older child.

I know. I answered that above your last post. It's 2/3.

Thanks semck83. Turns out that while I can't read, I am not actually going mad. I was fixated on your original question and over looked your revised question, where you made it clear that the question was about the probability of there being two boys.

Draug pointed this out to me about a page ago. But as I had been stating the question as I understood i I was curious to find out if my answer 2/3 was correct.

It was confusing to me to still be told by multiple people that my reasoning was wrong (despite the alternate premises).

Sorry spy, I probably should have realized some time ago that that's what you were referring to.

I agree with smeck on that one (and with your empirical results, but you've got to see that that writing out the possibilities, then circling the favorables is easier). With the assumption that at least one child is a boy you have the three possibilities as before, two of those possibilities have the youngest being a boy, either they're both boys or the oldest is a girl.

It was a great little puzzle semck83. It reminds me of the Monty Hall problem. Both questions are deceptive because think we perceive the answer too quickly. We have this 50/50 heuristic that we apply without properly considering the question.

Actually that is what I did originally philcore

1. Boy Boy - younger child boy
2. Boy Girl - younger child girl
3. Girl Boy - younger child boy

The excel was a sanity check because I was told I was wrong. But it turns out that I had a different interpretation of the question than everyone else. Hence the confusion. Sorry.

So we are agreed:

Knowledge he has a boy makes the odds he has another boy 2:1 against
Meeting one of the kids makes the odds of the other child being a boy even
Meeting one of the kids makes the odds of the second (i.e. younger) child being a boy is 2:1 in favor

100% agree.

Thinking about this again, I was able to confuse myself again by wording it this way:

I meet a boy named Tom. He says he has one sibling. There are 4 possibilities, he either has an older brother, an older sister, a younger brother or a younger sister. what is the probability his sibling is a boy? This is just a re wording of the original problem, yet by enumeration the possibilities, it now seems that the prob is 50% that there are 2 boys in the family, given the knowledge that they have at least one. Surely the fact that I gave the one a name doesn't change the odds?

Who can tell me where I went wrong? Here are the possibilities:
TB
TG
BT
GT

2 are favorable out of 4.

I think I figured out the answer, but I want to hear from others before I say.

That is 50%.

Yes, 50%. Tom's sex becomes irrelavent because he is removed from the equation as a known factor.