Calculating possibilities correctly

As these days seem to be devoted to raising public attention to breast cancer, I will write the rest of this blog in pink.

I read in another blog a posting about calculating possibilities correctly. I quote (my words are in normal type, the other blog is in italics):

How about “most women with breast cancer have a positive mammogram”? We’d likely agree. So what about “most women with a positive mammogram have breast cancer”? Not so, most women with a positive mammogram do not have breast cancer. Most diagnosticians don’t get this right when presented with the raw statistics – 1% probability that woman of 40 has breast cancer, 80% chance if she’s got it that the mammogram detects it and a 9.6% chance she’ll test positive if she hasn’t got it. The experts reckon that, based on these numbers, a woman with a positive mammogram has a 75% chance of having breast cancer.

They’re wrong, and not a bit wrong. They're massively wrong.

Putting the numbers a different way, “10 out of 1000 women at age forty who participate in routine screening have breast cancer [1%]. 8 out of every 10 women with breast cancer will get a positive mammogram [80%]. 95 out of 990 women without breast cancer will also get a positive mammogram [9.6%].”

Calculating this properly: out of 1000 women tested, there are 103 positive mammograms, of which 8 are true positives (breast cancer) and 95 are false positives. So the possibility of a woman with a positive mammogram having breast cancer is 8/103 or 7.8%.