Question

(1) According to the Journal of the American Medical Association, a 40-year old woman, who is symptom free, has a 0.4% (0.004) chance of having breast cancer. Suppose she undergoes a mammogram and receives an “abnormal” result. According to the National Cancer Institute, mammograms report “abnormal” 80% of the time when a woman does, in fact, have breast cancer and 10% of the time when no cancer is present.

(a) Given the "abnormal" mammogram result, what is the probability the woman has breast cancer?

Now, after receiving the “abnormal” result discussed above, she undergoes a biopsy. If a patient has breast cancer, the biopsy will correctly identify the cancer 95% of the time. If a patient does not have breast cancer, the biopsy will correctly identify the patient as not having cancer 90% of the time. Unfortunately, the biopsy reports that cancer is present.

(b) What is the probability that the woman has breast cancer given the "abnormal" mammogram and the biopsy report?

Answer 1

Let events

• c= a randomly selected 40 year old symptom free woman has breast cancer
• nc=a randomly selected 40 year old symptom free woman does not have breast cancer
• am=event that the mammogram result is abnormal
• nm=event that the mammogram result is normal
• ab=event that the biopsy result is abnormal
• nb = event that biopsy result is normal

We know the following probabilities

a) The marginal probability of getting an abnormal mammogram result is

Given the "abnormal" mammogram result (event am) , what is the probability the woman has breast cancer (event c)

ans: Given the "abnormal" mammogram result , the probability the woman has breast cancer is 0.0311

b) We know that Given the "abnormal" mammogram result , the probability the woman has no breast cancer is

We know the following probabilities regarding the biopsy results

the probability that the woman has breast cancer (event c) given the "abnormal" mammogram and the biopsy report (events am,ab)

ans: the probability that the woman has breast cancer given the "abnormal" mammogram and the biopsy report is 0.2338