Question 2. (15 Points)

Suppose that, in the course of a regular check-up, a doctor discovers that the patient has a potentially cancerous lesion. Most lesions are benign (non-cancerous), say 99%. The doctor orders an x-ray just in case. In laboratory tests on malignant (cancerous) lesions, the x-ray returns positive (cancer-affirming) results 79.2% of the time and negative results 20.8% of the time. In laboratory tests on benign lesions, the x-ray returns positive results only 9.6% of the time and negative results 90.4% of the time.

1.(5 Points) The patient's x-ray comes back positive. What is the probability that the patient has cancer?

2.(6 Points) Suppose that the doctor calculates the probability that the patient has cancer without regard to the base rate of cancer in the population. That is, the doctor uses Bayes' rule but assumes that cancerous and non-cancerous lesions are equally likely. What mistaken conclusion will the doctor draw from the test?

3.(4 Points) Explain why it is important in these situations to have hospital procedures that require additional tests to be performed before a patient undergoes treatment.