That's really nice. Thanks for the link. Bayes' theorem is a lot less complicated than its enthusiasts make it seem, especially Yudkowsky. If you understand conditional probability and if you know how to check your work in algebra then you already know Bayes' theorem.
This post makes me wish I had access to the web when I went to school. I would probably have launched a collaborative lets-explain-the-math-book-together site or something. Or maybe I would just have played World of Warcraft all day long.
What would you have done (or: what did you do if you're not an old f*rt such as me) if you went to school today, with all the technology available?
Probably the most helpful resource online for me academically has been Wikipedia. I have had TAs and professors who recommended reading specific Wikipedia articles to brush up on related topics, especially in math classes.
There is a demand for this kind of thing outside of just high-school. I'm not in a position to go back to high-school or university, but I'd like (and would pay) to be able to "study along" with other people on a course/textbook.
(This is partly a lazyweb request - if this exists, someone tell me where!)
I wish authors of these sorts of articles put the correct solution first instead of the incorrect one. I'm more likely to remember the first proof I read and this sort of thing screws me up.
Other than that, I found it a really interesting read.
I will confess: I never memorized Bayes theorem. I just imagine rectangles getting chopped into overlapping pieces, and visually derive it every time I need to use it. I've found that this actually worked better for me than trying to apply a formula, since you're less likely to forget intuitions when you're thinking through a problem.
Pictures are, by far, my favorite way to explain Bayes theorem.
Monty Hall problem explained wrong. There is no "probability" of Monty opening any "goat door". Monty KNOWS which doors contain goats,and always opens one of them. You have a better chance choosing the remaining door Monty "owns" because he had a 2-out-of-three chance to begin with (you chose randomly and uniformly). He still has that chance. No information was added when he opened a door, because he can ALWAYS open a goat door.
It is explained correctly. There are two doors with goats, and he opens one of them randomly if you didn't pick one of them. Note that he explains it twice; the first time with the common logic error that he tends to make and the second correctly.
Whether Monty opens a door or not doesn't change your probability - the door you chose has 1/3 chance. Wave your hands all you want and show off bogus math if you like.
Ok, smart guys. Imagine Monty didn't have any "probability" of opening a "goat door": he just opens the 1st one (or only one if he only holds 1). It doesn't change anything about the problem; but that fake math quits working.
I almost believed you, until I ran through this version of the problem, and realized that, if I know Monty always opens the lower-numbered goat door, it actually gives a completely different result.
In particular, if I pick door 2, and Monty picks door 3, this _guarantees_ that the car is behind door 1 under your version of the problem. Meanwhile, if I pick door 2, and Monty picks door 1, it is now 50/50 whether I should switch or not, instead of 2/3 to 1/3. Try the math yourself and see.
So the reason the "fake math" quits working is that you've changed the problem; if you do the math using the method at the link, for your version of the problem, you _rightly_ get different results, and I believe they are correct.
