But aren't I ignoring the base rate in the actual numbers one? Or no?

The actual numbers in the problem were chosen in such a way to make the base rates obvious. Here is another version using real numbers where the base rates aren't quiet so obvious, see if you can get it right:

"1 out of every 100 women at age forty who participate in routine screening have breast cancer. 80 out of every 100 women with breast cancer will get positive mammographies. 96 out of every 1,000 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?"