A woman has two kids. One is a boy. What are the odds the other is a boy?

Scenario 1: The younger child is named Nathan (50%)
- 25% chance that the older child is a boy
- 25% chance that the older child is a girl
Scenario 2: The older child is named Nathan (50%)
- 25% chance that the younger child is a boy
- 25% chance that the younger child is a girl

50%.

With the original phrasing, it can be understood three ways.
1. The question is referring to a particular kid of the two, and states that it is a boy. Therefore, the one not mentioned can either be a boy or a girl => 50%.
(To elaborate, there are four possibilities. The 'one' mentioned in the problem is blue:
B/B
B/B
B/G
G/B
The odds that the other is a boy is 2/4 = 50%.)
2. The question states that exactly one kid of the two is a boy, meaning that the 'other' child must be a girl => 0%.
3. The question states that at least one kid is a boy. Likewise, it can also be understood that the woman has at least two kids in the first place => indeterminate.

This is fun.

Let's analyse each sentence:

A woman has two kids.
Should be easy enough. The woman does not have 1 kid or 3 kids, but 2 kids.

One is a boy.
This is rather cryptic:
First thing about the sentence is that we don't have any information about which kid it is, so it could be either the first or the second kid.
Now, the second thing about the sentence is that it states the gender of this kid. But was this gender picked out randomly, or with intention? Was the intention to state that one of these woman's kids are a boy, or was it to state that one of the kids have xx or xy gender?

What are the odds the other is a boy?
Easy enough. If given the condition in the sentence before this, what is the probability that the other kid is a boy? It also backs up that the woman has two kids, but this should be clear enough already.

If the kid was picked out randomly, that would mean that either child X or child Y was chosen on beforehand. This gives your four possibilities, and therefore 50% chance.

If the kid was picked out based on gender, that would mean that there's 33% chance, based of all the other pages.

3. The question states that at least one kid is a boy. Likewise, it can also be understood that the woman has at least two kids in the first place => indeterminate.

This isn't really indeterminate. Say we have N kids, and one of these is a boy. That gives us 2^N - 1 different combinations, as we have to remove the all-girl-possibility. Let us now assume that the "other" is taken randomly out of the group of remaining kids. If this group has n boys, then the chance you'll choose a boy is n/(N - 1). Summing up these different chances may be explained as the following function: