1. (Problem 13.10) Show that the statement P(A,B | C) = P(A | C) P(B | C) is equivalent to either of the statements P(A | B,C) = P(A | C) and P(B | A,C) = P(B | C).

2. (Problem 13.11) Suppose that you are given a bag containing n unbiased coins. You are told that n-1 of these coins are normal, with heads on one side and tails on the other, whereas one coin is a fake, with heads on both sides.

a. Suppose you reach into the bag, pick out a coin uniformly at random, flip it, and get a head. What is the (conditional) probability that the coin you choose is the fake coin?

b. Suppose you continue flipping the coin for a total of k times and see k heads. Now what is the conditional probability that the coin you picked is the fake coin.

c. Suppose you wanted to decide whether the chosen coin was fake by flipping it k times. The decision procedure returns FAKE if all k flips come up heads, otherwise it returns NORMAL. What is the (unconditional) probability that this procedure makes an error?

3. Consider a simple Bayesian belief network on two diseases among smokers.

In the above network, "T" refers to whether the person has tuberculosis, "C" refers to whether the person has lung cancer, "X" refers to whether the lung X-ray shows a problem, "S" refers to whether the skin test indicates tuberculosis.

Compute the following probabilities. Show all the steps of the computations.

a. P(X | C)

b. P(C | X)

c. P(C | X, T)

4. (Problem 14.3) Two astronomers in different parts of the world make measurements M₁ and M₂ of the number of stars N in some small region of the sky, using their telescopes. Normally, there is a small possibility e of error by up to one star in each direction. Each telescope can also (with a much smaller probability f) be badly out of focus (events F₁ and F₂), in which case the scientist will undercount by three or more stars (or, if N is less than 3, fail to detect any stars at all). Consider the three networks shown in Figure 14.19 (below).

a. Which of these Bayesian networks are correct (but not necessarily efficient) representations of the preceding information?

b. Which is the best network? Explain.

c. Write out a conditional distribution for P ( M₁ | N), for the case where N ∈ {1,2,3} and M₁ ∈ {0,1,2,3,4}. Each entry in the conditional distribution should be expressed as a function of the parameters e and/or f.

d. Suppose that M₁ = 1 and M₂ = 3. What are the possible numbers of stars if we assume no prior constraint on the values of N?

e. What is the most likely number of stars, given these observations? Explain how to compute this, or, if it is not possible to compute, explain what additional information is needed and how it would affect the result.