# Can someone explain the Boy Girl Paradox to me?

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It’s so counter-intuitive my head is going to explode.

Here’s the paradox for the uninitiated:If I say, “I have 2 kids, at least one of which is a girl.” What is the probability that my other kid is a girl? The answer is 33.33%.

>*Intuitively, most of us would think the answer is 50%. But it isn’t. I implore you to read more about the problem.*

Then, if I say, “I have 2 kids, at least one of which is a girl, whose name is Julie.” What is the probability that my other kid is a girl? The answer is 50%.

>*The bewildering thing is the elephant in the room. Obviously. How does giving her a name change the probability?*
>
>*Apparently, if I said, “I have 2 kids, at least one of which is a girl, whose name is …” The probability that the other kid is a girl* ***IS STILL 33.33%.*** *Until the name is uttered, the probability remains 33.33%. Mind-boggling.*

And now, if I say, “I have 2 kids, at least one of which is a girl, who was born on Tuesday.” What is the probability that my other kid is a girl? The answer is 13/27.

>*I give up.*

In: 4

There are four equally likely gender configurations of families that have two children: male/male (1), male/female (2), female/male (3), female/female (4). The statement that at least one is a girl eliminates family #1. So you’re picking randomly from the three other families. Only in family 4 is the other child a girl. So one in three odds.

This is a variant of the Monty Hall problem. You have an unknown probability and some information. So, if you just say “I have two children”, there are four options:
* two boys
* older girl, younger boy
* older boy, younger girl
* two girls.
When you add the fact that one is a girl, you have only eliminated the first possibility. However, that fact says nothing about which of the other three possibilities it is. In two of the three cases, the other child is a boy, so what’s left is a 1 in 3 chance that the other child is a girl.

Giving a name or a birthday adds more information. There are 27 possibilities for one child being born on a Tuesday, 13 of which are the pairing of two girls.

The name is even more unique, as there are a near infinite number of names the children could have. Technically the probability of a second girl is 49.99999…999%, but you can round to 50 because there are only so many people who would name their children the same thing.

*George Foreman has entered the chat*

As for the Julie part by saying one of the children is Julie you no longer have this distribution: MM MF FM FF. Instead you have Julie/m Julie/f OR m/Julie f/Julie.

By knowing that on child is not just a girl, but a specific girl you have a different distribution of possibilities.

For the scenario “I have 2 kids, at least one of which is a girl, whose name is Julie”, start again with the basic possibilities:

Boy/Boy. Boy/Girl. Girl/Boy. Girl/Girl.

Now, *the parent knows* which of the children (the older or the younger) is Julie. Let’s say that Julie is the younger child.

So now the possibilities become:

Boy/Boy. Boy/Julie. Girl/Boy. Girl/Julie.

There are only two possibilities in which Julie is the younger child, and of those possibilities the chance of both children being girls is 1 in 2, or 50%.

We could also have assumed that Julie is the older child, and the result is the same.

This “paradox” depends on a linguistic trick where by naming the child you are changing one interpretation of the phrase’s meaning and considering a different situation.

Consider the first question: “I have 2 kids, at least one of which is a girl. What is the probability that my other kid is a girl?” There are four different possible ways the children could be born:

Girl and Girl

Girl and Boy

Boy and Girl

Boy and Boy

We can eliminate the last option from consideration because one of them isn’t a girl, meaning we only have the first three. Of those three only the first has the other child being a girl, so the probability is 33.33%

It is important to note that the situation of “Girl and Boy” is being counted as distinct to “Boy and Girl” even though they both equate to there being one boy and one girl. This is because while the end result is the same the probability of having both boys or both girls is not the same as one of each. It is this difference which the “paradox” is exploiting with ambiguous phrasing.

Consider the second question: “I have 2 kids, at least one of which is a girl, whose name is Julie.”

In this situation it is being interpreted that we are picking a child from an existing pair of children, which combines the “Boy and Girl” and “Girl and Boy” possibilities. So instead we have these options:

Girl and Julie

Boy and Julie

Boy and Boy

Again we can eliminate the last option from consideration because we know one isn’t Julie, meaning we only have two options left. Therefore it is now 50% that the other child is a girl.

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However, I would argue this is an improper twisting of linguistics and probability. We already established that the chances of having both a boy and a girl is equal to the chances of having both children the same sex. Therefore we would expect that there would be twice as many families out there with Julie and a boy compared to Julie and a girl, even though there are just a pair of options. Just because there are two options doesn’t mean they are equally probable.