[spadros]

Yes, I found this one easy. Was surprised my data management intuition came back after all these years since school. There’s really only three options:

- boy - boy

- boy - girl

- girl - girl

So it must be 1/3 chance. If you’re looking at permutations in order, that’s a different question.

[AIPedant]

This intuition is wrong even if turned out to get the right answer. The three unordered options do not have equal probabilities, boy+girl is twice as likely to occur as boy+boy and girl+girl.

To get the right answer you must be careful about conditional probabilities (or draw out the sample space explicitly). The crux of the issue is that you are told extra information, which changes your estimate of the probability.

(This question as written is very easy to misinterpret. The Monty Hall problem, which illustrates the same thing, is better since the sample selection is much more carefully explained.)

[taeric]

Oddly, this is a part I'm sticking with on this problem.

Specifically, if you know that one is a girl, then the unordered options seem like they are back on equal footing? That is, it isn't twice as likely if you know that one ordering can't happen? (Or, stated differently, you don't know which version of two girls you are looking at.)

So, for this one, you know that either the youngest is a girl (so, girl-boy is not possible) or that the oldest is a girl (so boy-girl is out). That puts you back to the rest of the possibilities. Boy-boy is out, sure, as you have a girl. But every other path remains? So, you have one of (boy-girl(known), girl-girl(known), girl(known)-boy, girl(known)-girl). Which drops you back to 50/50?

[AIPedant]

Like others in the thread have said, the question could have been phrased more precisely. Technically you are misreading it but in an annoying and trivial way.

What the problem is really saying is this:

1) You have a large collection of families with two kids of varying genders.

2) You draw one of them at random. At this point, your only estimate of P(2 girls) is 0.25.

3) Someone tells you that the family you drew has at least one girl.

4) This extra information changes your probability estimate because the possibility of two boys has been ruled out; the naive 1/4 estimate is refined to 1/3.

The way you are interpreting it is this:

1) You have a large collection of families with two kids, at least one of whom is a girl.

2) Then the probability that the other child is a girl is clearly 50%.

As a reminder this is how the original post phrased the question:

  Here's the problem: a family has two children. You're told that at least one of them is a girl. What's the probability both are girls?

This is just too vague and admits both interpretations, they needed to be more specific about where the family "came from." That's why Monty Hall is a better illustration: it starts with you explicitly choosing a door at random. Here the family has been chosen at random from the pool of families with two children, but that's totally unclear.

[taeric]

The annoying thing is this sits with my teaching fine, it is more my intuition that is failing to withstand trying to break it. :(

So, in the original: "a family has two children. You're told at least one of them is a girl." What are the possible states? Well, assume first born is the girl, then you have 50% that the next is a girl. Then, assume that the first born was a boy, then there is no chance and the second born is the girl that you know of. So, at 50/50 on those chances, you have 50% chance of having a 50% chance, or a 50% chance of it being 0. I can't see how to combine those to get 1/3. :(

And the Monty Hall explicitly covers the case that a decision is made on which door is shown to you. I don't see any similar framing to this problem. Yes, the total states are GB, BG, GG, but only if you treat GG in such a way that either BG or GB was not a possible state. (That is, using G for girl that you know of, and g for unknown, then possible states are GB, Gg, gG, BG. There is no version of Bg or gB that is possible, so to treat those as equal strikes me as problematic.)

[AIPedant]

Where you're getting confused is by trying to combine state space determination and probability determination at the same time (this is also why the problem is so similar to Monty Hall). The state space is shifting when you say "assume X, then the probability of Y." You are going back and forth between using and not using the information to decide arbitrarily that some probabilities are 50% and others are 0%, which leads to an invalid conclusion.

Specifically: it is not true that the firstborn has a 50-50 chance of being a girl, given you were told that the family has at least one girl. The firstborn has a 2/3rds chance of being a girl. This is the heart of your confusion.

In a broader sense there is an entire class of confusing conditional probability problems like this. Events which are causally independent in reality (e.g. gender of a child, which door Monty Hall hid the car behind) fail to be probabilistically independent when you have extra information. Yet these probability games are contrived in a way that our intuition takes over and we use our causal understanding even when a better probabilistic understanding gives you a better answer.

[taeric]

But in the monty hall, the "discloser" is limited by my choice/observation. It is literally part of the framing. In this framing, there is no limit based on my choice.

Consider, if you tell me that the Smiths have at least 1 girl, and I meet their daughter but you haven't met either, I have no way of knowing if I met the 1 girl or not. I could ask you, but you would just say, "I don't know, could be her. Could be the other kid. I just know they have at least 1 girl."

This is very different from the monty hall case, where the announcer knows what is behind all doors.

Similar. But different. I was using coins as an example elsewhere. If I flip a dime and a quarter, and tell you that one of them is heads, do you have increased chance of knowing if either particular one is heads? This is more liar's dice than it is monty hall.

[kgwgk]

> 4) This extra information changes your probability estimate because the possibility of two boys has been ruled out; the naive 1/4 estimate is refined to 1/3.

That’s not correct in general.

It’s only correct if you assume that “3) Someone tells you that the family you drew has at least one girl.” was equally likely to happen whether or not there were two girls.

That’s a quite strong assumption.

One can make different assumptions and get answers different from 1/3. For example, 1/2.

[AIPedant]

Unlike the other misreading, I think you are splitting hairs about something boring and irritating. Rephrase the problem instead to "you have a reliable device that can tell whether the family has at least one daughter but doesn't tell you how many."

[kgwgk]

> boring and irritating

Ok, AIPedant.

I understand that you may find irritating that someone points out that the original problem is ill-posed and “the answer” depends on how we decide to “rephrase” it.

However, it doesn’t seem boring in the context of a discussion of how the problem is not well-posed and additional assumptions are required to get an answer.