# 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

## 1 Answer

No, this is not true in real life: There are good theoretical reasons, and ample empirical data, suggesting that the sex of siblings is positively correlated (very weakly), such that a family that already has a girl as their first child is more likely to have another girl than a boy. Early empirical analysis of this matter can be found in Harris and Gunstad (1930) and James (1975). Essentially this occurs because the sex of each child gives some statistical information on underlying causal characteristics, and this induces (very weak) positive correlation in the sex of siblings (on this latter point, see e.g., O'Neill 2009).

• The papers by Harris and Gunstad (1930) and James (1975) were brilliant. They convinced, it is not just by chance that someone gets a boy/girl as 2nd child but something different than "chance" is going on. So, in your first sentence you say its not true and then you with your data and proved its true. Why say "not true" in first place ? – Arnuld Nov 19 '18 at 4:17
• @Arnuld: Perhaps I misunderstood your question, but I took you to be asking whether it is really true that the sex of each child is independent with a fifty-fifty chance of a boy or girl (since that is the underlying assumption of the toy problem in your question). The papers cited here are saying that these events are not statistically independent; the sex of children from the same parents is positively correlated. This would mean that the toy problem in your question approximates --but does not exactly follow-- births in real life. – Ben Nov 19 '18 at 4:26
• @Arnuld: In other words, I am saying that if a couple's first child is a girl, then their second child will be a girl with probability (slightly) higher than 50%. – Ben Nov 19 '18 at 4:27
• Let me see if I get your point. Yeah, Probability (as in above toy-problem :) ) can not be exactly true for every couple out in the world. Based on data from excellent papers, our guess a.k.a probability seems to be holding good for population on whole than to a specific couple. I completely understand whole vs specific thing. Is this what you are saying ? – Arnuld Nov 19 '18 at 4:34
• Not really --- the issue is not a distinction between specific couples versus the whole population. For a specific couple, the data (and accompanying theoretical statistical arguments) tell us to expect slight positive correlation between the sex of children. The population is just the aggregate of all child-bearing couples, and in the population this manifests in higher proportions of boys born in families that already have more boys, and vice versa (see e.g., Table of proportions of boys in James 1975). – Ben Nov 19 '18 at 21:20