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by | Oct 19, 2020

Imagine two new families move into your neighborhood, the Smiths and the Joneses, and both have two children. You learn that the Smith’s oldest child is a girl but not the gender of the younger child and that one of the Jones’s children is a girl, but you don’t know whether she is the older or younger sibling. What is the probability for each family that the second child is also a girl?

Shockingly, the odds are different. The chances of the second Smith child being a girl is 1/2 while the chance of the other Jones child being a girl is 1/3rd. How is that possible! Here’s how:

For families with two children, here’s the possible combinations:

Once you know that the Smiths have an older girl that knocks out the two scenarios where there is an older boy. The only two options are the second child is a boy or a girl. It’s 50-50 or a 1/2 chance the second child is a girl.

For the Joneses, however, you only know that one of the children is a girl, but not the order. That means three possible scenarios exist: Girl/Girl, Boy/Girl, or Girl/Boy. In only one of these scenarios does the Jones girl have a sister: in the Girl/Girl scenario. Thus, the chances that the Jones’ other child is a girl is one in three.

Thus, if you walk through the math, it’s clear that knowing the birth order of the one child who is a girl affects the probability that the other child is a girl. This is similar to the Monty Hall Problem. This counterintuitive answer is due to the fact that knowing one child is a girl means there cannot be two boys.

BUT WAIT! That’s crazy. The chance of each sibling being either gender is 50-50. If you knew nothing about the gender of either child, the chances of either sibling being a boy is 50-50 and are independent of each other. So, it is crazy to think that chances of a child being a girl goes from 1/2 to 1/3 just because we know the gender of the sibling but not the birth order.

This is an apparent paradox – the two equally correct answers seem to be in conflict. But they actually aren’t in conflict. It’s a matter of framing and shows how learning new information can change probabilities (just like in the Monty Hall problem). If you ask “for any family with two children, what are the chances that they have one boy and one girl” the answer is 50% (and a 25% there are two boys and 25% that there are two girls). The Jones situation is different, it’s asking “for any family with two children where you know one is a girl, what is the chance the other is a girl?” That answer is 1/3rd.

Ok, here’s a few other interesting things related to this. First, actually, the chances of having a boy vs. a girl aren’t exactly 50-50. There are 1.05 males born for every female. Here’s an IFOD about why: Why Do Females Live Longer Than Males? Also, more attractive couples are more likely to have a girl as their first-born. IFOD on that is here: Attractive Couples Have More Daughters.

1. The Uncertainty principle of the quantum theory may help perhaps…a specific outcome is a result of the collapse of the wave function out of the mental and biological superposition of cells, neurons and a quantum mental scenario that leaps into both the past and the future in order to determine that specific outcome….

2. I remember having a philosophical discussion with a stats teacher about the independence of events. Any single time I flip a coin it has a 50/50 chance of coming up heads. But the odds of me getting several heads in a row start getting dramatically small the more times I flip. So if I’m on my 10th flip and the first 9 have been heads, then independently by 10th flip is still a 50% chance of getting heads. But given the the odds of flipping 10 heads in a row is 1/1024, it sure feels like the universe wants me to flip a tails on that one. 🙂

3. I thought about this a lot and what’s crazy is someone could say: “You go over to meet your new neighbors and see their daughter playing outside. You know they have two children. What is the probability that the second child is also a girl?” The answer is 1/2, even though it is similar to the Jones situation in that you do not know whether the daughter is the older child or younger child. I think the trick here is a semantic one. The question about the Jones from the IFOD is asked in a way that path dependency is not relevant for the girls, but it is relevant for the boys. Trippy.

• Exactly. You explained it much better than I did. Here’s from a paper on this paradox: “To simplify matters, let us return to Gardner’s problem: ‘Among all two-child families for which at least one child is a boy, for what fraction of these families are there two boys?’ We have found above that the answer is P = 1/3. Now consider this scenario: you are strolling on Dun Laoghaire pier and meet an old school-chum, Pat, whom you have not seen since your youth. He is accompanied by a boy, and introduces him thus: ‘This is Jack, one of my two children’. What are the chances that his other child is a boy? The answer is P = 1/2 ; Pat’s family has not been pre-selected from those having at least one boy.”

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