[peakok]

You are right about the question ("Announcing the gender of one child does not magically alter the gender of the other child"), but wrong about the statistics, even in your second formulation the probability is 2/3, not 1/2. This problem, the way Jeff has formulated it, has nothing to do with the Monty Hall problem. But still, the end probability is the same : 2/3 (that's the only common point). Tell me if there is a mistake :

The question is : What are the odds that the person has a boy and a girl, if we already know that one of the child is a girl ? (If we agree on the question, then we must agree on the following probabilities).

Possibilities are :

1/ boy / boy

2/ boy / girl

3/ girl / boy

4/ girl / girl

Since one of them is a girl, we must remove possibility number 1. That leaves us with 3 possiblities, and 2 of them have a boy. Probability : 2/3

(edited for correction, we search the probability of having a boy, not a girl :p)

[timr]

Ordering is irrelevant here. Options 2 and 3 are identical.

[peakok]

This is precisely the first subtility you encounter when you learn probabilities : they are not identical. Paul is right that the formulation of the question doesn't refer to the Monty Hall problem at all, this isn't the same algorithm. But in this case, the probability turns out to be the same. That's the real confusion in Jeff's article.

I mean, it's not even my own deduction, it's what you are taught when you learn probabilities. It's a basic and core paradigm, and I'll digg it up from Wikipedia if somebody still doubts it :)

[timr]

See my post above. You can take ordering into account if you'd like, but the result is the same, given the parameters of the problem.

[randallsquared]

Options 2 and 3 each still have the same probability weight as each of options 1 and 4.

[timr]

No, they don't.

Again, ordering is irrelevant in this problem. We want to know only the probability that there will be a boy/girl pair, not the probability that the boy/girl pair was born in a particular order.

But the same result can be obtained when taking ordering into account -- the key observation is that if you subdivide options 2 and 3 to account for sibling ordering, then you also must subdivide options 1 and 4 to account for sibling order, resulting in 6 total possibilities, of which 2 are M/F sibling pairs, 2 are M/M pairs, and 2 are F/F pairs:

1) M/m

2) m/M

3) M/F

4) F/M

5) F/f

6) f/F

Given the knowledge that one sibling is female, you then exclude 2 of the 6 possibilities (the m/M and M/m pairs), to obtain 2/4 = 50% probability that the pair of siblings is of mixed sex.

The mistake you're making is that you're including ordering on the mixed-sex pairs, but not including ordering on the same-sex pairs.

[peakok]

If you include m and f in the universe of possibilities, then you must add the following combinations as well :

7) m/f

8) f/m

You are not allowed to skip arbitrarily some of the combinations of your universe (wich is now [M, F, m, f]). Probability : 2/3 =]

[timr]

Don't be silly -- do you think I invented a couple of new sexes by adding lower-case letters? You're just getting thrown by the notation. I could have written the options as:

1) M/M

2) M/M

3) M/F

4) F/M

5) F/F

6) F/F

but I thought that was confusing, so I introduced a symbol to more clearly illustrate the differences between the ordering of the same-sex options.

[peakok]

"Somebody is wrong on the internet". I'm wasting my time and this is my last answer. If you haven't noticed, I only use strict mathematical arguments and I invite you to do the same if you intend to answer. Pure and clear maths please, no litteral arguments about the sexes or god knows, this is the only field where we can verify it.

Lacking elementary probabilities knowledge isn't as dramatic as refusing to learn it, please teach yourself now since nobody will look at this thread again.

Here is my last point, and you can ask any teacher of formal logic, probabilities or maths to verify it :

For universe [M,F], the table of possibilities is :

MM

MF

FM

FF

And that's it. I'm sorry but the table you just made up doesn't exist at all, please go ask one of your teacher about it. If you still believe you are right, and you can prove it, you just discovered a new field in mathematics and probabilities, congratulations.