CMPSCI 240: Reasoning About Uncertainty

Third Midterm Exam

David Mix Barrington

17 April 2009

Directions:

• Answer the problems on the exam pages.
• There are six problems for 100 total points. Actual scale is A=90, C=62.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• When the answer to a question is a number, you may give your answer in the form of an expression using arithmetic operations, powers, falling powers, or the factorial function. Probabilities may be given as either fractions or decimals.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 30 points
  Q6: 30 points
  Total: 100 points

• Question 1 (10): True or false with justification: If I believe that there is a 42% chance that the Red Sox will win tonight's game, and I can bet on their winning at three-to-two odds (with no service charge), my expected return (expected money won minus expected money lost) is positive.

• Question 2 (10): True or false with justification: I think that there is a 1/3 probability that it will rain tomorrow. I consult my local TV station, which predicts that it will rain, but I know that their predictions are correct only 60% of the time. After considering their prediction, I still believe that the probability of rain is less than 50%.

• Question 3 (10): True or false with justification: Let PR be an object that produces a boolean value whenever its next method is run. Suppose that any even number of consecutive booleans produced by PR consist of exactly half true and exactly half false. Then PR may be used in place of a true generator of uniform random booleans in any application.

• Question 4 (10): True or false with justification: A particular test for Disease C has a false positive rate of 10% and a false negative rate of 30%. If a patient tests positive with this test, I can conclude that there is a 60% chance that she has Disease C.

• Question 5 (30): I am going to throw two fair, independent, ordinary six-sided dice. Let M be the maximum of the two numbers thrown, and let T be the total of the two numbers thrown. Let D be the event that the numbers on the two dice are the same ("doubles").
  • (a,5) For each number i in the set {1, 2, 3, 4, 5, 6}, compute Pr(M = i).
  • (b,5) Compute the conditional probabilities Pr(D | M = 6) and Pr(D | M ≠ 6).
  • (c,5) Compute the prior odds O(M = 6) and the posterior odds O(M = 6 | D).
  • (d,5) Compute the conditional probabilities Pr(T = 7 | M = 6) and Pr(T = 7 | M ≠ 6).
  • (e,5) Compute the posterior odds O(M = 6 | T = 7).
  • (f,5) Multiply the prior odds O(M = 6) by both the likelihood ratios you used in parts (c) and (e). Is the result the posterior odds O(M = 6 | D ∧ (T = 7))? Explain your answer.

• Question 6 (30): Last year I saw 25 movies in the theatre, 15 of which I liked. Three professional critics, named Paula, Randy, and Simon, also rated each of these movies "thumbs up" or "thumbs down". I would like to use a Naive Bayes Classifier to help me use these critics' opinions of future movies to predict whether I will like them. Summarizing their opinions:
  • Paula liked 12 of the 15 movies that I liked, and liked 8 of the 10 movies that I did not like.
  • Randy liked 10 of the movies that I liked, and liked 4 of the movies that I did not like.
  • Simon liked 3 of the movies that I liked, and liked 5 of the movies that I did not like.

  Here are your questions:

  • (a,5) Suppose I am considering a new movie that comes from the same distribution as the 25 movies from last year. What should be my prior odds and prior probability that I will like the movie?
  • (b,10) What should be my posterior odds and probability that I will like the new movie if all three critics like it? What if all three critics hate it?
  • (c,10) Given the three critics, there are eight possible sets of their opinions on a new movie. What set or sets of critics' opinions make it most likely that I will like the new movie, and what are the odds and probability in that case? What set or sets will make is least likely, and what are the odds and probability in that case?
  • (d,5) I would classify 10 of the 25 movies from last year as science fiction, and I liked 8 of those 10. When I evaluate the probability that I will like a new movie, I would like to take into account whether it is a science fiction movie. I could either add this feature to my Naive Bayes Classifier, or I could run two separate Naive Bayes Classifiers, one for the SF movies and one for the non-SF movies. What are some reasons why one of these ideas would be better? Do you need information that I have not given you?

Last modified 22 April 2009