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January 08, 2004

Why Is Probability So Hard?

I am going to pull out some familiar chestnuts here. If you have not heard these puzzles before, then good, you may find them interesting. If you have heard them before, even better.

First, let's start with something obvious. If you toss a fair coin once, does the result tell you anything about the result of the next toss? No.

Which leads to...

The Twin Paradox
(Yeah, I know that the name is already taken).

A mother pushing a pair of non-identical twins in a baby carriage calls you over. She tells you that one of them is a boy.

What is the probability that the other child is a boy?

And this is generally where the instructor gets to condemn everyone who answers 1/2.

Here are the possible sex groupings of the twins:
1. BB
2. BG
3. GB
4. GG

Only number 4 can be ruled out. Therefore the chance of there being two boys is 1/3, not 1/2.

The Game Show Paradox

On a game show there are three curtains. Behind two of the curtains are goats. Behind the other curtain is a Ferrari. The contestant has to choose one of the curtains to open, and he gets whatever is behind that curtain.

So Bob the contestant picks a curtain to open. Before it is opened, however, Pat the host opens another curtain, showing the goat behind it. He then gives Bob the opportunity to switch curtains. Should Bob switch?

And this is generally where the instructor gets to condemn anyone who says it doesn't matter.

When Bob picked the first curtain, there was a 1/3 chance that the Ferrari was behind it, and a 2/3 chance that it was behind another curtain. So it is obvious that if he stays with the curtain he is at there will only be a 1/3 chance of winning the Ferrari. But if he switches, he will have a 2/3 chance.

No Good At Probability?

Nearly everyone's gut feeling was 1/2 and 'doesn't matter' respectively. This proves we're no good at probability, right? Not so fast.

There is actually a good reason for our gut feelings. And it is the same reason in each case. Let's go back to the twins.

Scenario A: Imagine that you had a great many mothers and their twins together in a room. You collect all of the mothers who could can truthfully say "one is a boy." This is 3/4 of the mothers. Then you check the sex of either one of children and record your results.

Scenario B: Imagine instead the same situation, except this time you ask the mother to tell you the sex of either one of the children. When the mother says "boy" true for 1/2 of the mothers you then check the sex of either one of the children.

So here is the secret to the twin paradox: We solved the twin paradox as A, but B was how we heard it.

B is a narrative in other words a sequence of events. That is how we tell stories. The twin paradox was presented as a narrative.

For want of a better term, A can be referred to as a description of a system. That is how we usually set up mathematics problems. We describe a system and then solve it. The twin paradox was solved in this manner.

The game show paradox is exactly the same.

Scenario A: Imagine a game show where the host knows where all of the prizes are. When a contestant picks one curtain, the game show host will always pick out one of the other curtains where a goat is.

Scenario B: Imagine a game show where the host knows nothing about the prizes. When a contestant picks one curtain, the game show host will pick a curtain at random and show what is behind it.

We solved the game show paradox as A, but B was how we heard it.

So this is not a problem of being good at probability or not. Our minds solve these problems just fine. It is a matter of language being too inexact to specify to our mind just which problem it should be working on.

Posted by Thrasymachus at 11:12 PM