1. 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 this person will have a positive hemoccult test. If a person does not have colorectal cancer, the probability is 3 percent that this person 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.

2. Suppose we have three random Variables A, B and C. Suppose that A and C are binary (True/False) and that B can take on three values (1,2,3). These variables are related in the following Bayesian Network (sorry for the crude arrows):

    A
/ \
v   v
B   C

Suppose also that the following (conditional) probabilities govern:

P(A=True)=0.4

P(B=1|A=True)=0.6 P(B=2|A=True)=0.1

P(B=1|A=False)=0.2 P(B=2|A=False)=0.7

P(C=True|A=True)=0.8

P(C=True|A=False)=0.9

Note that in each case I expect you to be able to figure out the “missing” probability.

A) Compute the Joint distribution for A, B and C.

Use this table to compute:

B) P(A=True|C=True)

C) P(B=3|C=False)

D) P(A=True|B=2,C=True)

3. (Adapted from Pearl (1988) ). Three prisoners, A, B, and C, are locked in their cells. It is common knowledge that one of them is to be executed the next day and the others are to be pardoned. Only the Governor knows which one will be executed. Prisoner A asks the guard a favor: “Please ask the Governor who will be executed, and then take a message to one of my friends B or C to let him know that he will be pardoned in the morning.” The Guard agrees, and comes back later and tells A that he gave the pardon message to B. What are A's chances of being executed, given this information? Show how you come to your answer mathematically. Failing that try running a little simulation.