Here's how I think about these problems. You know that the woman tested positive. There are two kinds of women who test positive: women who actually have breast cancer and correctly test positive (the true positives), and women who don't have breast cancer but mistakenly test positive (the false positives). How big are these two groups?
First, the true positives: women who actually have breast cancer and correctly test positive. 1% of women actually have breast cancer, and 80% of them test positive, so 0.8% of women are in that group (.01 x .8).
Then, the false positives: women who don't have breast cancer but mistakenly test positive. 99% of women don't have breast cancer, and 9.6% of them mistakenly test positive, so 9.5% of women are in that group (.99 x .096).
What you care about is the relative size of the two groups. For every 0.8 women who have cancer and test positive, 9.5 women don't have cancer but still test positive. That's a 0.8:9.5 ratio; if you want to turn it into a percentage it's 0.8/10.3 = 7.8% of the women who test positive are ones that actually have cancer.
So instead of using the whole formula, I think through the problem and do three simple calculations along the way. If you only need to calculate a rough estimate, you can do this even quicker with less calculating. Glancing at the numbers, about 1% of women are in the true positive group, and about 10% of women are in the false positive group, so about 1/11 of women who test positive have cancer (9%). That's pretty close to the actual answer of 7.8%.
This is probably going to sound utterly ridiculous, but I have a sad confession.
I've read Yudkowsky's post on Bayes' Theorem (http://yudkowsky.net/rational/bayes) five times. I've written down the equation. Tried to formulate an answer.
I still don't understand it. That being said, I've lived my entire life under the false mentality that maths is boring and painful, and it's just recently I've tried to actually understand the concepts I learn in school, and not just temporarily memorize them for the next exam.
Here's the problem, on Yudkowsky's post:
"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?"
When Eliezer changes the percentages to real numbers:
"100 out of 10,000 women at age forty who participate in routine screening have breast cancer. 80 of every 100 women with breast cancer will get a positive mammography. 950 out of 9,900 women without breast cancer will also get a positive mammography. If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?"
When I see this equation, I can properly make the answer come out to 7.8 percent. I do this, by taking the 80 women, and dividing 80 women by the 80 women plus the 950 women, so 80/80+950 (or 80/1030=.078). So I get 7.8%, which should be the right answer.
But when I try to do the same with percentages, it all gets sort of screwy. I take the 80 percent of women (.8) divided by that same 80 percent (.8) plus 9.5 percent of women without cancer who test postive for it (.095). So I get .8/.8+.095=89%.
I feel like I'm making a really, really stupid error. But I just don't know what it is. >_>