Comment by klodolph | Hacker News Reader

Comment by klodolph | Hacker News Reader

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You're oversimplifying.

The problem is the interpretation of the question in precise terms. We are given some background information and then told the observation of a particular random variable. In the given article, the random variable is interpreted as "yes/no: at least one child is a boy born on a Tuesday". Rambling about "order" does not change the fact that the article is correct, given the definition of the random variable that is observed.

The problem is that this interpretation is only correct if you imagine that you are always given the observation of the same random variable no matter what the genders or birthdays of the children actually are. For example, if I had two girls born on Wednesday, I would then tell you "no child is a boy born on a Tuesday". That's an unlikely scenario which is what makes it legitimately surprising.

It's like the Monty Hall problem. What if the host picks the door in advance? What if the host only opens a door with a goat under the condition that you picked the door with the prize? What if the host only opens a door with a goat under the condition that you picked the wrong door?

The article has problems but "wrong" is not one of them.

> so hey, fuck you guys too

Hacker News is a terrible place to visit if you want to take the voting system personally. Don't make this about you. It's about the post.

Wait, explain it to me like I'm 10 years old.

As I see it, there are three options - either it's a boy-boy pair, a girl-girl pair or a boy-girl pair. Now, since OP introduced the age variable, let's give the kids names to make things easier to understand. Let's say that the boys are named John and Jack, the girls Jill and Judy and if it's a boy-girl pair then they're named Jack and Jill.

Now if it's a boy-girl pair, we have exactly two options: either Jack is older than Jill or Jill is older than Jack. This is shown in the first picture.

However, we also have two options for boy-boy and girl-girl pairs: Either Jack is older than John or John is older than Jack. Either Jill is older than Judy or Judy is older than Jill. This is not shown.

Since we know that one of them is a boy, we can dismiss both girl-girl pairs. Now we're left with four options. In 50% of these options the other child is a boy.

What am I missing?

> As I see it, there are three options - either it's a boy-boy pair, a girl-girl pair or a boy-girl pair.

Yes, but those three options are not equally likely.

Let's say the parents decide that their firstborn will be named Jack if male and Jill if female, and the second child will be John if male and Judy if female. Now you see that there are four possibilities: Jack+John, Jack+Judy, Jill+John, Jill+Judy. The last one is eliminated when you know that there is one boy, which gives 1/3 chance that it's the first.

The reason age was introduced was simply was a way of differentiating the two children (giving them identity). You introduced names to differentiate the two children, which is another perfectly good way to do it.

But then you _also_ introduced age. Because you now have two different ways of labelling the children, you've introduced a new variable, and that changes the probability.

A better way to look at it is with two unisex names: Drew and Sam. This completely replaces the age thing as a way of identifying them, so forget all about age.

Our possibilities are:

  Drew: boy, Sam: boy
  Drew: boy, Sam: girl
  Drew: girl, Sam: boy
  Drew: girl, Sam: girl

We know Drew and Sam can't both be girls, because at least one is a boy, so that leaves:

  Drew: boy, Sam: boy
  Drew: boy, Sam: girl
  Drew: girl, Sam: boy

There's a 1/3 chance that both Drew and Sam are boys.

Ok, yes, I get it now :). Thanks for the explanation.

I'm not, you're overcomplicating! X is a boy. What is the chance that Y is also a boy? 1/2 or 1/3? Non-rhetorical, answer the question as posed.

> answer the question as posed

You can't give a correct answer to an incomplete question. Sit down and define the problem and if you come up with a different answer it's because you made a different assumption about the definition of the observed random variable.

Let's suppose there are four universes.

* Two boys (25% chance)

* Two girls (25% chance)

* Boy and girl (50% chance)

If we assume that the observed random variable has a preimage that contains all possibilities where the statement is true, then we come to the 1/3 conclusion, which is counterintuitive but true. This is the position of the article. I have come to the same conclusion without using order, but I have used the same definition of the observed random variable.

Tell me what you think the random variable is.

We could assume that the random variable has a preimage which only contains the case where both children are boys. Bam, probability 100%, not 1/2 or 1/3. Or we can assome that the random variable has a preimage which only contains the case where one child is of each gender. Bam. Probability 0%. So any answer between 0% and 100% is defensible.

God, I knew this was coming. Since you won't accept my strategy of building up to the complete question (because it is in fact your formulation which is incomplete, and thus I, like you, will not respond to it), let me try this instead:

You misunderstand Monty Hall. I would know; I learned about it by being asked the question and getting it right, not by reading someone else's explanation (which leaves room for misinterpretation). The key to Monty Hall is none of what you stated (or rather, rhetorically asked); it's that the host must pick a door that is both a) bad and b) not yours. If your door is good, then he can leave either, but if your door is bad, then he must leave the good one. So, two possibilities: the door he left was either bad or good, BUT the rule driving his choice wasn't 50/50, so the odds of it being bad or good aren't 50/50, so neither are the odds of yours, even though there are only two left. Take a minute, think about it.

Now ask yourself which "door" (read child) the father told you was a boy. There's the gotcha. Under your/the author's formulation of the question, he doesn't remove a door in the BB universe, but he does in the BG universe. There are two BG's because the table acknowledges that he can tell you which of the two is definitely a boy, and thus he has two options for doing so BUT HE DOESN'T, HE HAS ONE BUT THERE ARE TWO BG UNIVERSES, SO IT LOOKS LIKE TWO.

As mentioned in other replies, it's clear now that you are interpreting the question in a manner other than the usual interpretation, and then being incredibly rude about anyone who disagrees with you.

I have in my hand a device that lets me call up on demand most of the knowledge of the world, and I use it to look at pictures of cats, and to argue with strangers.

Deleting the second option ...

> Since you won't accept my strategy of building up to the complete question (because it is in fact your formulation which is incomplete, and thus I, like you, will not respond to it),

No, his formulation is complete and inevitably leads to the correct answer 1/3. You are constantly avoiding a complete formulation of the problem by diverting to a muddled discussion of an entirely different problem. This is NOT about Monty Hall.

Or you're trolling. In which case you are doing it masterfully.

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