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According to data from the Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute, white men in the U.S. have about an 11% chance of being diagnosed with prostate cancer during their lives, black men have about a 16% chance, and all other (non-white and non-black) men have approximately an 8% chance.

  1. 6 points Suppose three unrelated black men are randomly selected from the U.S. population. What is the probability that all three will be diagnosed with prostate cancer at some point during their lives?
  2. 6 points Suppose three unrelated black men are randomly selected from the U.S. population. What is the probability that at least one of them will be diagnosed with prostate cancer at some point during his life?
  3. 6 points Suppose three unrelated black men are randomly selected from the U.S. population. What is the probability that none of them will be diagnosed with prostate cancer at some point during their lives?

In the U.S., approximately 73% of men are white, 13% are black, and 14% are neither black nor white.

  1. 9 points What is the probability that a randomly selected man from the U.S. population will be diagnosed with prostate cancer during his life?
  2. 9 points Suppose a randomly selected man from the U.S. population is (or will be or has been) diagnosed with prostate cancer. What is the probability that he is black?
  3. 9 points Are race and being diagnosed with prostate cancer independent events? Explain.

Colorectal cancer (CRC) affects more than 1 million people each year, and has an overall poor survival rate. It is important to diagnose CRC early in order to engage in treatment to ensure best outcomes, but CRC is often asymptomatic or has mild symptoms in its earliest stages. The 5-year survival rate for stage 1 and 2 cancer is 90%, but the survival rate for stage 3 cancer is 71% and for stage 4 cancer is 14%. Thus, it is important for older adults to be regularly screened for CRC.

Traditionally, CRC is diagnosed via colonoscopy, which is an invasive technique in which an endoscope is inserted into the colon to look for evidence of cancer. Colonoscopy is considered the “gold standard” of colon cancer screening, but due to its invasive nature, is undesirable for many patients. A non-invasive screening test using stool samples was developed and is known as the Cologuard test. In a large study of 10,000 patients of average risk for CRC, Cologuard demonstrated 92% sensitivity and 94% specificity in detecting CRC (Imperiale et al., NEJM 2014) among adults aged 50-84.

  1. 12 points Suppose the prevalence of colon cancer in a given population was 1% and you were to administer the Cologuard test to a randomly selected human in this population and it came back positive. What is the probability that they have CRC?
  2. 13 points In a population that has a CRC prevalence of 1%, what proportion of positive test results are false positives? Cologuard recommends that any positive Cologuard test be followed up with a colonoscopy for confirmation. Using your calculation, explain why this would or would not be a good recommendation.
  3. 15 points What prevalence in the population would you need for a test given to a random person to have a positive predictive value of more than 50%?
  4. 15 points Cologuard’s advertising materials state that it is “a highly effective test that may help you rule out CRC.” Is this a reasonable statement to make? Explain, and support your answer with any data and/or calculations you deem necessary.

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