Paradoxes of Probability and Other Statistical Strangeness

In this problem, as you've described it, we're enumerating "ways to assign gender and birth-day-of-week." We can do this because there are a countable number of "days of the week" (so we can map to the integers: 1-6) AND there is also a surjective function of [child] -> [day of the week they were born]. Am I right so far?

Now let's replace the set [1-6] with another countable set that also maintains the surjective function. We could say "day in the lunar cycle" (so ~27 options), or better "day of the year" (366 options), for example. Do we now need to consider the 23662366 ways to assign gender and birth-day-of-the-year? Take it further with whatever you want: "birth weight in milligrams" or "number of freckles" (as I previously suggested). All countable things that meet the surjective requirement.

This is starting to smell funny, right? So let's take a look at the math.

You say there are 2727 ways to configure day+gender, assuming independence for kid 1 (k1) and kid 2 (k2). This represents: (k1 gender options * k1 day of week options) * (k2 gender options * k2 day of week options). Right? I'm with you so far. Then you say "Of these possibilities, 27 are situations where one kid is a Tuesday boy." Hold up.

We are given two pieces of information: that one of the kids is a boy, and that particular boy was born on a Tuesday. Let's say the boy is k1 (this is an assignment of enumeration, not of "who came first;" just like Sunday = 1 does not mean that any kid born on a Sunday was born before every kid born on Monday = 2). So now the k1 options are [11] (boy, tuesday), and the total number of options are: [11] * [27] = 14. Of those 14, 7 are girl options. And we're back to a straight 50%.

So yes, I dispute the 27 number. It seems like it is arrived at by 2127, minus one for an apparent duplicate. But the 212*7 represents maintaining gender non-specificity for Tuesday boy, which should be incorrect, no?

> You have stated by fiat that certain things are irrelevant to certain other things...

Yes, but that's what "independent" means, right? You also stated that you're assuming these two things are independent, hence equiprobability. But independence is defined by P(A) = P(A|B). The probability of A is completely unaffected by B. Yet the outcome you arrive at is that P(A) IS affected by B, so the math presented is internally inconsistent.

What am I missing here? I'm fascinated by the uncertainty around this little problem.

If K1 is a girl and K2 is a boy born on Tuesday, this still counts as the family (Mr. Jones, if you like) having a boy born on Tuesday. There are 27 cases that count as the family having a boy born on Tuesday, all equiprobable. And out of those, 14 also count as the family having a girl.

As for your noting that we can split these cases even more finely, so that there's no distinguished end-all, be-all partitioning of cases, sure, you can do that. What I'm really saying is this:

1/2 of two-child families have their elder child being a boy. 1/2 of two-child families have their elder child being a girl. [On conventional idealizations for these problems. You surely do not dispute this, yes? You may not care about this number, but you don't dispute it, right?]

In each of those subgroups, 1/7 of families have their elder child born on Sunday, 1/7 have their elder child born on Monday, etc. [Do you dispute this?]

In each of THOSE subgroups, 1/2 of families have their younger child a boy, and 1/2 have their younger child a girl. [Any dispute?]

And in each of THOSE subgroups, 1/7 of families have their younger child born on Sunday, 1/7 of families have their younger child born on Monday, etc. [Any dispute?]

And some amount of those have low birth weight, some have high birthweight, some have 5 freckles, etc., but we needn't figure out those numbers.

So now I've carved the world up into 2 * 7 * 2 * 7 groups, based on gender and birth-date-of-week for older and younger child. We can carve the world up into groups in different ways also, more finely or more coarsely or just differently. But making the four conventional assumptions we just made, the 2 * 7 * 2 * 7 grouping based on gender and birth-date-of-week for older and younger child is such that each particular such group takes up 1/2 * 1/7 * 1/2 * 1/7 of all families; these are all equifrequent groups.

And that having been done, we find that in 27 of these groups, there is at least one boy born on a Tuesday. In 13, the elder child is a boy born on Tuesday but not the younger child; in 13, the younger child is a boy born on Tuesday but not the elder child; in 1, both children are boys born on Tuesday.

But the question was not intended to be about a specific boy. The question was intended to be "Out of families that have a boy born on Tuesday (meaning at least one boy born on Tuesday), what proportion have a girl?". Any family with at least one boy born on Tuesday counts as having "a boy born on Tuesday", and even families with two boys born on Tuesday count, with no particular of their two boys given any distinguished status.

Perhaps you read the question differently; that, then, is a problem with the phrasing of the question for communicating to you its intent. But when it was asked "What is the probability Mr. Jones has a girl, given that he has a boy born on Tuesday?", what the author indeed intended this to mean, and would be generally taken in the conventional language of probability to mean, was "Out of families that have at least one boy born on Tuesday, what proportion have a girl?".

And we find that, out of the 27 equally sized groups of families that have at least one boy born on Tuesday, 14 of them have a girl, so that the answer to this question becomes 14/27.