

Probability as Space - jballanc
http://thebitfarm.blogspot.com/2009/01/probability-as-space.html

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psyklic
The real point of confusion is that people don't realize that BG/GB are two
separate states. Like many of the recent blog articles, this one also doesn't
clearly explain why you need to consider both BG and GB. Consider:

"A person has two kids, and his first-born is a boy. What is the probability
the other is a boy?"

There are two possibilities, each equally likely -- (first-born, second-born):
(B,B) and (B,G). Hence, 1/2.

"A person has two kids, and one is a boy. What is the probability the other is
a boy?"

In other words: "The first-born is a boy AND/OR the second-born is a boy."

There are three possibilities, each equally likely -- (first-born, second-
born): (B,B), (B,G), and (G,B). Hence, 1/3.

~~~
psyklic
To be more clear: When hearing this problem, some people imagine -- (one-kid,
other-kid): (B,G) and (B,B), hence 1/2.

Others imagine -- (first-born, second-born): (B,B), (B,G), (G,B), hence 1/3.

Someone needs to write a blog post explaining why the latter method is
superior, not just a blog post that automatically enumerates the four states.

People need a REASON for not combining BG/GB, and considering the children as
(first-born, second-born) provides much of that reason, since it is a logical
way of ordering children. A (one-child, other-child) ordering appears a bit
absurd, even to the layperson.

~~~
jballanc
The reason is that the two children are different dimensions in the
probability space. Both (one-kid, other-kid), and (first-born, second-born)
yield the same space, and both will give a 1/3 answer.

~~~
psyklic
People interpret "one kid is a boy" as "first-born is a boy" in my example
above (i.e. they selectively choose which kid is the "one kid" -- the one
which is a boy), which yields 1/2.

People seem to be good at enumerating the "probability space" quite well. They
just don't understand why the dimensions are what they are.

