# Third Midterm Exam

### 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.