Understanding Bayesian Theorem

Eliezer S. Yudkowsky says:

Maybe you see the theorem, and you understand the theorem, and you can use the theorem, but you can't understand why your friends and/or research colleagues seem to think it's the secret of the universe. Maybe your friends are all wearing Bayes' Theorem T-shirts, and you're feeling left out. [...] What matters is that Bayes is cool, and if you don't know Bayes, you aren't cool.

His first example in a nutshell (by M. H. Herman):

Here's a story problem about a situation that doctors often encounter: 1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

Only around 15% of doctors get it right (Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995; and many other studies.)…On the story problem above, most doctors estimate the probability to be between 70% and 80%, which is wildly incorrect…The correct answer is 7.8%, obtained as follows: Out of 10,000 women, 100 have breast cancer; 80 of those 100 have positive mammographies. From the same 10,000 women, 9,900 will not have breast cancer and of those 9,900 women, 950 will also get positive mammographies. This makes the total number of women with positive mammographies 950+80 or 1,030. Of those 1,030 women with positive mammographies, 80 will have cancer. Expressed as a proportion, this is 80/1,030 or 0.07767 or 7.8%”

I modeled this as BN with GeNie and came to the right result:
yes = 7.8% genie displays it rounded to 8%
I learned hereby:

• I don't understand Bayes well yet
• I don't need to understand Bayes to model a Bayesian Network successfully