Here's my solution, and a comment on the implications in this application.
In the following, let C denote cancer, M+ denote a positive mammogram, and C^ and M+^ donote not having cancer and not having a positive mammogram, respectively.
The probability that a woman in the age range has breast cancer is P(C) = 0.01.
The probability that a woman's mammogram is positive given that she has breast cancer is P(M+ | C) = 0.8.
The probability that a woman's mammogram is positive given that she doesn't have breast cancer is P(M+ | C^) = 0.096.
We are required to find the probability that a woman has breast cancer given her mammogram is positive, P(C | M+).
From conditional probability:
P(M+ | C)P(C) + P(M+ | C^)P(C^) = P(M+)
so P(M+) = (.8 * .01)+(.096*(1-.01)) = 0.1030
Bayes' theorem states that:
P(M+ | C) = P(C | M+)P(M+) / P(C)
We re-arrange to find:
P(C | M+) = P(M+ | C)P(C) / P(M+) = .0777.
So, the probability of a woman having breast cancer given that she has a positive mammogram is just under 10%.
Assuming these probabilities are correct, one might be tempted to say that this is unacceptable. However, it is worth thinking whether it would be preferable for fewer positive mammograms to be reported (given it's hard to know which ones are true positives and which are false). Doing so would result in fewer (false) scares---but would also result in more cancers being missed. It is also worth asking whether a better method for detecting breast cancer even exists (hint: not yet).