Doesn't this rest on the simple ambiguity in the phrasing?
> I have two children and one is a son born on a Tuesday.
If by that is meant:
> I have two children. Here is some information about one of them: son, born on Tuesday.
Then the probability of the other child being a son is 1/2.
If on the other hand we mean:
> I have two children. One or more is a son. Exactly one of them was born on a Tuesday.
Then we get the 13/27 probability.
In fact it doesn't seem reasonable at all to assume that only one was born on Tuesday, while at least one is a son. One single interpretation of 'one of them' must be applied to both the gender and day of birth. Otherwise we're picking and choosing our interpretation on a whim.
edit: Colin appears* to think that what I've said here is incorrect, and I'd like to know why. I'm not a maths/stats person at all so am very keen to be re-educated on this matter.
* based on his now-deleted reply to ars which said "no, it doesn't, and no, you're not"
Important Edit Two:
If I'm reading this right, I think the defender of the 13/27 solution would say:
No. We don't discount the possibility of both being tuesday-boys (TB), we just adjust the calculation so that it doesn't count (eldest=TB, youngest=TB) and (youngest=TB, eldest=TB) as two separate possibilities.
To which I respond:
Right, so it's not down to ambiguity. But shouldn't you also discount every other symmetrical pair such as (eldest=TB, youngest=WB) and (youngest=TB, eldest=WB) and thus return the odds to 1/2? Or does that not return the odds to 1/2?
No, you shouldn't "also discount every other symmetrical pair", for exactly the same reason as there's a 1/36 chance of rolling double 6, but 2/36 chance of rolling a six and a one. It's all to do with labellings, and it's the most common source of error[1] in statistics.
[1] By "error" I mean calculations that then don't agree with the experimental results.
Doesn't this mean that the more information we gain about the boy, the less likely it makes it that his sibling is a brother?
More likely, but only if that information being true was a precondition for knowing about the boy in the first place. Take the following scenarios, assuming Alice knows Bob has exactly two children.
Alice: Do you have a son?
Bob: Yes
Alice: Pick one of your sons, and tell me the day of the week he was born
Bob: Sunday
Here the day of week provides no additional information because Bob will always have an answer (like in Monty Hall, where Monty will always reveal a losing door), so the probability that Bob has two boys is 1/3.
Alice: Do you have a son who was born on a Sunday?
Bob: Yes
Here having a son isn't enough; he also has to satisfy a condition that occurs with only 1/7 probability. Bob is more likely to be able to answer yes if he has two sons and thus two chances to satisfy that condition.
> I have two children. One or more is a son. Exactly one of them was born on a Tuesday.
I'm not sure that's what you are supposed to infer.
Looking at your earlier statement:
> I have two children. Here is some information about one of them: son, born on Tuesday.
There are two ways to interpret this.
(1) I am a man pulled at random from the set of [families with two children of indeterminate gender]. Here is some information about one of them: son, born on Tuesday.
(2) I am a man pulled at random from the set of [families with two children of indeterminate gender, one of whom was born on a Tuesday]. Here is some information about one of them: son, born on Tuesday.
We're not selecting from the same initial set in each case - set (2) is more restrictive. A difference in probability is maybe not surprising.
Still, I totally agree with you that it's a bit of a jump to conclude that the man is referring to scenario (2) in which the birthday information is used to narrow the initial set while the gender information is used to determine the probability. Just seems like a trick question to me.
To expand on the numbers a bit more... In scenario (2) the possible combinations are:
(Combo A) G G (0/49 at least one boy TB)
(Combo B) B G (7/49 at least one boy TB)
(Combo C) G B (7/49 at least one boy TB)
(Combo D) B B (13/49 at least one boy TB)
If the man has one boy, then combo A does not apply and the probabilty he has two boys must be 13 / (7 + 7 + 13) == 13/27.
Yeah, I was very confused what exactly the "paradox" was at first, too, and why the expected interpretation should be the one it was. Here's how I finally see it:
If I say, "I have two cars, one's a 1994 Porsche 911", then I think it's reasonable to interpret that statement to mean the other car is not also a '1994 Porsche 911', but doesn't speak at all to its Porsche-ness, 1994-ness, or 911-ness. Rather, I'm wrapping them all up in a package, and saying the other is not this exact combination of characteristics.
Similarly, if I say, "I have two children. One's a son born on Tuesday," I think I now understand that I'd probably interpret that to mean the other child is not a son born on Tuesday. It doesn't speak to the gender or day of birth beyond that.
With that knowledge, that the other child is not a (son AND Tuesday-born), that's when the 13/27 arises.
The ambiguity isn't in the phrasing of the father, we understand what he says about his family, it
is in the probability universe that surrounds his statement. The scenario doesn't give one; we don't have a model for the probability of people saying anything at all unprompted. We have a simplified model for the probability distribution of genders, and one for how prompts and replies modify an already known probability model, but here there is nothing to anchor any reasoning of this kind.
> I have two children and one is a son born on a Tuesday.
If by that is meant:
> I have two children. Here is some information about one of them: son, born on Tuesday.
Then the probability of the other child being a son is 1/2.
If on the other hand we mean:
> I have two children. One or more is a son. Exactly one of them was born on a Tuesday.
Then we get the 13/27 probability.
In fact it doesn't seem reasonable at all to assume that only one was born on Tuesday, while at least one is a son. One single interpretation of 'one of them' must be applied to both the gender and day of birth. Otherwise we're picking and choosing our interpretation on a whim.
edit: Colin appears* to think that what I've said here is incorrect, and I'd like to know why. I'm not a maths/stats person at all so am very keen to be re-educated on this matter.
* based on his now-deleted reply to ars which said "no, it doesn't, and no, you're not"
Important Edit Two:
If I'm reading this right, I think the defender of the 13/27 solution would say:
No. We don't discount the possibility of both being tuesday-boys (TB), we just adjust the calculation so that it doesn't count (eldest=TB, youngest=TB) and (youngest=TB, eldest=TB) as two separate possibilities.
To which I respond:
Right, so it's not down to ambiguity. But shouldn't you also discount every other symmetrical pair such as (eldest=TB, youngest=WB) and (youngest=TB, eldest=WB) and thus return the odds to 1/2? Or does that not return the odds to 1/2?