Of all 40-year-old women, 1% have breast cancer. If a woman has breast cancer, a mammogram will give a positive indication for cancer 90% of the time. If a woman does not have breast cancer, a mammogram will give a positive indication for cancer 9% of the time. If a 40-year-old woman’s mammogram gives a positive indication for cancer, what is the probability that she has cancer?

To find the probability that a 40-year-old woman has breast cancer given that her mammogram gives a positive indication for cancer, we can use Bayes' theorem.

Let's define the events:
A: Having breast cancer
B: Mammogram gives a positive indication for cancer

We are given the following probabilities:
P(A) = 0.01 (1% of 40-year-old women have breast cancer)
P(B|A) = 0.90 (if a woman has breast cancer, the mammogram gives a positive indication 90% of the time)
P(B|¬A) = 0.09 (if a woman does not have breast cancer, the mammogram gives a positive indication 9% of the time)

We want to find: P(A|B) (the probability of having breast cancer given a positive mammogram).

According to Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)

To find P(B), we can analyze the two possible scenarios:
1. The woman has breast cancer (A)
2. The woman does not have breast cancer (¬A)

P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A)
= 0.90 * 0.01 + 0.09 * (1 - 0.01)

Let's calculate P(B):
P(B) = (0.90 * 0.01) + (0.09 * 0.99)
= 0.009 + 0.0891
≈ 0.0981

Now, we can use Bayes' theorem to find P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.90 * 0.01) / 0.0981

Let's calculate P(A|B):
P(A|B) = (0.90 * 0.01) / 0.0981
= 0.009 / 0.0981
≈ 0.0917

Therefore, the probability that a 40-year-old woman has breast cancer given a positive indication on the mammogram is approximately 0.0917 or 9.17%.