Question

The fecal occult blood test, widely used both in physicians’ offices and at home to screen patients for colon and rectal cancer, examines a patient’s stool sample for blood, a condition indicating that cancer may be present. A recent study funded by the National Cancer Institute found that of 15,000 people tested on an annual basis, 10% were found to have blood in their stools. Only these 10% underwent further testing, including colonoscopy, the insertion of an optical-fiber tube through the rectum in order to inspect the colon and rectum visually for direct indications of cancer. Only 2.5% of those having colonoscopy actually had cancer. Additional information in the study suggests that, of the patients who had the fecal occult blood test, approximately 5 out of 1,000 tested negative (no blood in the stool) but eventually did develop cancer.

a) The study results have led some medical researchers to agree with the American Cancer Society’s long-standing recommendation that all U.S. residents over 50 years of age be tested annually. On the other hand, many researchers claim the costs of such screening, including the cost of follow-up testing on 10% of the population, far exceeds its value. Assume that the test can be performed for as little as $10 per person, that colonoscopy costs $750 on average, and that about 60 million people in the United States are over age 50. What is the expected cost (including follow-up colonoscopy) of implementing a policy of screening everyone over age 50? What is the expected number of people who must undergo colonoscopy? What is the expected number of people who must undergo colonoscopy only to find that they do not have cancer after all?

Answer 1

Probability of patients undergoing colonoscopy = 10 % = 0.1

Cost of test = $10

Cost of colonoscopy = $750

Expected cost (including follow-up colonoscopy) of implementing a policy of screening everyone over age 50

= 0.9 * 10 + 0.1 * (10 + 750) = $85

Expected number of people who must undergo colonoscopy = 0.1 * 60 million = 6 million

Probability that those having colonoscopy actually had cancer = 2.5% = 0.025

Probability that those having colonoscopy do not had cancer = 1 -  0.025 = 0.975

Expected number of people who must undergo colonoscopy only to find that they do not have cancer after all

= 6 million * 0.975 = 5.85 million