Another unintuitive probability (to me anyway) is the size of a group of people you need to have before it's more likely than not that 2 of them will share the same birthday
There's another very counter-intuitive one, but it requires a bit of a build-up:
Say there's a village that only has families who have 2 children each. Say the probability of any particular child of being a boy or a girl is 50/50.
First, there's a slightly-counter-intuitive question:
* Given a family where at least one of the two children is a boy, what are the chances that the other child is a girl?
Now, the super-counter-intuitive continuation, assuming you solve the above correctly.
* Given that no two children in the village have the same name, and one of the children in some family is named Joseph, what are the odds that the other child is a girl? (The answer is different!)
For the 1st question, you are drawing only from families with boys (excludes families with 2 girls). Only a 1/3rd of picked families would have two boys, so 2/3rds have the girl as the other child.
For the 2nd question, if you are asking about whether Joseph's sibling is a girl, I think more information is needed.
How is the child named Joseph picked? If we assume that a random boy (regardless of family) is chosen to be named Joseph, 1/2 of boys have male siblings and 1/2 have female siblings (the boy-boy situation is double-counted if choosing a random boy), so it would be 1/2.
If we named Joseph by choosing a family with at least one boy and naming one of their boys "Joseph" randomly, you get the same 2/3rds as the 1st problem.
I don't see how knowing one of the children is named "joseph" gives us any more information than knowing one of them is a boy [1]. So I would be curious to see your logic for why the two have different answers.
[1] Unless you're leaving math-problem-world and want us to consider the probability that a child named 'joseph' is female.
Let's say we have a village with just 4 families, 1 for each possible case. The names of the boys are b0,b1,b2,b3 and the names of the girls are g0,g1,g2,g3 (as no two names are the same):
b0, b1
b2, g0
g1, b3
g2, g3
If we draw a random family that has at least one boy, we get all the families except the last one. And 2/3 of those have a girl in them, as opposed to 1/3rd that doesn't. So its 66% to find a girl.
However, if I ask you about a particular boy, I could ask about b0, b1, b2 or b3.
b0: sibling is b1
b1: sibling is b0
b2: sibling is g0
b3: sibling is g1
So for half of the possible names I choose, the sibling is a boy, and for half it is a girl.
Assuming the name was chosen at random from the existing names of the boys in the village, that makes the chance 50% to find a sibling girl again.
