I was surprised that nobody quizzed me when I wrote that a single individual could have two different sets of DNA. Anyway, for our purposes here we can assume that babies come in two flavours, male and female, and that statistically these appear in equal numbers.
Now lets get the easy, boring bit out of the way;
PART 1 - I have two children. One of them is a boy. What are the odds that the other is also a boy?
Lay your cards on the table.
radman:
...that statistically these appear in equal numbers.
But they don't.
PART 1 - I have two children. One of them is a boy. What are the odds that the other is also a boy?
If your wife typically eats a big breakfast about 55%.
@CodingBadly I agree, but here we have rules. There are boy babies and girl babies born statistically at the same rate. So I have two children one is a boy what are the odds that the other is also a boy?
Come on this is the boring part, boy/girl lets move on.
I don't think it matters. If I throw a coin and it comes up heads, what are the odds that the next throw comes up heads?
Nick, Nick you are a "Brattain Member", come on, you are up there with Dr Spock, we want logical, correct, computer programmer thinking not wooly, nonsense. This is the easy part.
PART 1 - I have two children. One of them is a boy. What are the odds that the other is also a boy?
Are you really saying the odds are 50/50 ?
Nick - I would add, good on you for playing the game and taking a position.
P.S. would you like to take a bet on this $)
Oh I see. You are asking what are the odds, given you already have the children? Not, what would the next one be?
Nick you are a "Brattain Member" ...
That just means I make a lot of posts, not that I know what I am talking about. 
50% as the birth of children is independent.
The other way of reasoning can be
there are 4 possible combinations
BG
BB
GG
GB
one is a boy so GG is removed, the list of possible pairs is
BG
BB
GB
one is a boy so the other is
G
B
G
that would imply 33%
PART 1 - I have two children. One of them is a boy. What are the odds that the other is also a boy?
@robtillaart is correct the odds of my other child being a boy are 1:3 - give the man a coconut
Note that this question would have a different answer if you said the eldest one is a boy...
Then the chance of the youngest being a boy is ...
nope
you seek to imply that the birth of one child has an influence on the birth of the next
leaving aside genetic tendencies (our family only ever has boys, twins, kittens)
the odds for the 2nd child are the same as the odds for the first
on contrary, in the original question there was no order information, in the variation there is.
Due to the order information the reasoning would be:
There are 4 possible combinations
BG
BB
GG
GB
The oldest is a boy so GG AND GB is removed, the list of possible pairs is
BG
BB
This leaves a 50% chance ...
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
ok so after 4 children what are the odds
by your logic you can't have more than 4
and even after three (if by then you have BBG)
there is still no guarantee that the fourth will complete the set
radman:
I have two children.
mmcp42:
ok so after 4 children what are the odds ...
Undefined.
radman:
P.S. would you like to take a bet on this $)
You would need to test that with loads of mothers/kids to get an accurate result, If... you have a load of nice good looking moms to choose from and you're willing to raise the kids, I may be interested.... 
PART 1 - I have two children. One of them is a boy. What are the odds that the other is also a boy?
@robtillaart correctly answered 1:3 or 33% (he also jumped the gun a bit on the next part)
So now onto the interesting bit;
PART 2 - I have two children. One of them, the thinnest, is a boy. What are the odds that the other is also a boy?
Cards on the table folks.
You probably want to say same as case 1
You probably want to say same as case 1
No I want you to tell me if the fact that this time the boy is thin changes the odds or not.
Place your bet.
Odds unchanged
Messieurs, dames - the first bet has been placed that skinny children don't change the odds.
Get your money on the table.