At a party you met someone. Here’s part of the small talk:

“Do you have any kids?”
“Yes, two.”
“Boys? Girls?”
“One of them is a girl.”

What are the odds that the other child is a boy?

I still have a hard time believing that it’s NOT 50-50.

Coding Horror asked the question and gave the answer as new year’s present. However, the “unfinished game” he gave as an example for the answer only confused more people, because the order (head then tail, or tail then head)is intuitively significant in the unfinished game, whereas order is intuitively UN-significant in the gender question.

I think it’s easier to understand it by asking a different question: If someone tells you s/he has two kids, and the older one is a girl…

The probability that the other one is a boy is a cold, hard 50%.

The answer is the same for this question: If someone tells you s/he has two kids, and the younger one is a girl…

It’s like seeing a little girl in a party, and someone besides her said “hi, this is my daughter Katie, she’s the older (or younger) sister.” Of course Katie either has a brother or a sister with 50-50 chance.

Now here’s the real tricky part, and that’s where the confusion comes from (I was confused again as I was writing this): in the same scenario, the parent said “hi, this is Katie, one of my two kids.”

Katie could have a younger or older brother, or a younger or older sister. So the odd IS STILL 50-50.

But that is NOT the original question.

The original question corresponds to this scenario instead:

At a party, you saw 2 girls and a boy playing together. Then someone said “one of those girls is mine, and her only sibling is also there”.

The key is you don’t know which girl the parent is talking about. His/her two kids can be girl A and girl B, girl A and boy, or girl B and boy. That’s why the odds for a boy is 2/3.

If instead s/he points to girl A and said “that’s one of my two kids”, then the answer is again 50%, because that eliminates one possibility.

It’s the same as this: if you saw 2 boys and a girl and someone said “my daughter is playing over there, and she has a sibling”. 50% boy.

My illustration isn’t perfect, though, as the answer depends on the group of kids you see. You get a different answer with a different composition. 3 girls and a boy: 50%. 2 girls and 2 boys: 80%.

In the original question, the parent is talking abstractly.

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