Assignment 3: Uncertainty Homework

GOAL: Understand what Bayes rule does, experience how powerful it is to represent real-world problems, and to explore use of this rule in a counter-intuitive problem.
  1. Problem 13.6 parts a, b, and c from the book.
  2. This problem comes from G. Gigerenzer, "Calculated Risks: How To Know When Numbers Deceive You", Simon and Schuster Press, 2002.  Give the answer and show how you obtain the results using Bayes rule.
To diagnose colorectal cancer, the hemoccult test --- among others --- is conducted to detect occult blood in the stool.  This test is used from a particular age on, but also in routine screening for early detection of colorectal cancer.  Imagine you conduct a screening using a hemoccult test in a certain region.  For symptom-free people over 50 years old who participate in screening using the hemoccult test, the following information is available for this region.

The probability that one of these people has colorectal cancer is 0.3 percent.  If a person has colorectal cancer, the probability is 50 percent that he [or she] will have a positive hemoccult test.  If a person does not have colorectal cancer, the probability is 3 percent that he [or she] will still have a positive hemoccult test.  Imagine a person (over age 50, no symptoms) who has a positive hemoccult test in your screening.  What is the probability that this person actually has colorectal cancer.
  1. This problem is from the same source, but is a variant of the previous problem.  Give the answer and show how you obtain your results.  You don't have to use Bayes rule, but you may if you want to.
To diagnose colorectal cancer, the hemoccult test --- among others --- is conducted to detect occult blood in the stool.  This test is used from a particular age on, but also in routine screening for early detection of colorectal cancer.  Imagine you conduct a screening using a hemoccult test in a certain region.  For symptom-free people over 50 years old who participate in screening using the hemoccult test, the following information is available for this region.

Thirty out of every 10,000 people have colorectal cancer. Of these 30 people with colorectal cancer, 15 will have a positive hemooccult test.  Of the remaining 9,970 people without colorectal cancer, 300 will still have a positive hemoccult test.  Imagine a sample of people (over age 50, no symptoms) who have a positive hemoocult test in your screening.  How many of these people actually have colorectal cancer?
  1. Problem 13.16 from the book.  The key to answering this problem correctly is to apply Bayes rule using the likelihood.  Note that we know something about how the information is obtained, and this knowledge impacts our final probabilities. (Hint, most people struggle to find the correct answer to this problem using Bayes rule.  If you can't figure it out using Bayes rule, try writing a simple program in MATLAB to estimate the result by running a bunch of simulations.)
  2. Complete the MATLAB Tutorial.  Write one paragraph summarizing the key points and a second paragraph identifying "muddy points", that is, things that you are struggling to understand.