Conditional Probability and its correctness

I am watching this YouTube video on conditional probability example by a university professor. He gives 2 examples:

• What is the probability of $$2$$ children being girls if we were told at least one of them is a girl (answer = $$1/3$$)

• What is the probability of $$2$$ children being girls if we were told the oldest one is a girl (answer = $$/1/2$$)

I understood both examples, he explained quite well. My question is: is this really true in life ? e.g. Let's say a couple has their first child,

1. If first child was a girl then they have 50% chance of 2nd child will be girl
2. If first child was a boy then they have 33% chance of 2nd child will be girl

Does this really happen ? Have we checked the data related to births to verify its correctness or we are sure formula works because we believe probability has been proved useful over 3 centuries or we just do it this way because it is mathematical formula ?

or is it just a mathematical formula ?

• I am unable to see how the first bullet applies to the numbered statements. It's a fact, though, that biology will not serve to demonstrate the incorrectness or lack thereof of any general mathematical principle or definition: that's not how science or math work. – whuber Nov 18 '18 at 18:00
• based on "If first child was a boy then they have 33% chance of 2nd child will be girl", maybe you should go back to that video again. – user158565 Nov 19 '18 at 3:05
• @user158565 33% = 1/3 – Arnuld Nov 19 '18 at 4:19
• @whuber you shook up my beliefs about Math and Science but then you leave without saying anything to read more on how science and math work :( – Arnuld Nov 19 '18 at 4:21
• at least one of them is a girl $\ne$ first child was a boy – user158565 Nov 19 '18 at 5:18