CMPSCI 240: Reasoning About Uncertainty

Third Midterm Exam

David Mix Barrington

18 November 2009

Directions:

• Answer the problems on the exam pages.
• There are six problems for 100 total points. Actual scale was A=90, C=60.
• 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. If you give your answer using the "choose" notation, also give it using only operations on this list. In addition, if your answer is a non-negative integer less than or equal to 100, you must compute the number for full credit.

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

• The first two true-false questions involve the following method of generating random bits. I take a standard 52-card deck, with 26 red cards and 26 black cards, shuffle it so that every ordering of the cards is equally likely, then deal five cards in order without replacement. My first random bit is 0 if the first card is red and 1 if it is black, the second bit is 0 if the second card is red and 1 if it is black, and the third, fourth, and fifth bits depend in the same way on the third, fourth, and fifth cards.

• Question 1 (10): True or false with justification: Each of these five random bits has probability 0.5 of equaling 0.

• Question 2 (10): True or false with justification: The probability that all five bits are 0 is strictly less than 1/32.

• Question 3 (10): True or false with justification: Suppose I choose a four-digit decimal number at random, with every number from 0000 through 9999 being equally likely. I then square this number and look at the low-order digit (the "units digit", which is the one furthest to the right). Then this digit is equally likely to be any of the digits from 0 through 9.

• Question 4 (10): True or false with justification: Suppose we train the Naive Bayes Classifier of Programming Project #3 (the original one, with single-letter features) on any 10 American and any 10 Russian cities. Then if we ask it to classify those same 20 cities as test data, it will classify all of them correctly.

• Question 5 (30): A fingerprint analyst has been asked whether any fingerprints from a crime scene match those of a particular suspect. She identifies six features of the suspect's prints, and determines that (a) a print from the suspect will have each of these features with probability 70%, independently of each other, and (b) a print from a randomly chosen person other than the suspect will have each of the features with probability 10%, independently of each other.
  • (a,10) The first print has all six of the features. Without the fingerprint evidence, the police estimate a 1% chance that this print belongs to the suspect. They ask the analyst whether she can now conclude, based on the six features, that there is a greater than 99% chance that it belongs to the suspect. Given the independence assumptions, can she do so? Justify your answer.
  • (b,10) It is clear that if a print has all six of the features, this makes it more likely to be from the suspect, and if it has none, this makes it less likely. How many features must she observe on a given print before the fingerprint evidence makes it more likely than it was before the fingerprint evidence was considered? (She looks for all six, so a print that has four of them also does not have the other two, for example.)
  • (c,10) Now suppose that the analyst chose her six features from a set of 50 possible features, and that her sample print from the suspect did not have any of the other 44. In the analysis above, she did not take account of whether the crime scene prints had any of those other 44 features. Describe how she could do so, assuming that these features are independent of the first six and of each other. What information about these features would she need?

• Question 6 (30): In last Sunday's football game, the Patriots led the Colts 34-28 with just over two minutes to go and had the ball on their own 28-yard line with fourth down and two to go. (If you are not familiar with American football, the following description of the situation should still be clear -- make sure to ask questions during the exam if it is not.) Patriot coach Bill Belichick had two choices -- he could punt or go for a first down. If he punted, the Colts would get the ball in their own territory and have some probability p of scoring a touchdown and winning the game. (Because the Patriots led by six, if the Colts did not score a touchdown then the Patriots would win. Throughout, we are ignoring a number of low-probability events that could have happened.) If he went for a first down, this would succeed with some probability q, If it succeeded, the Patriots would definitely win the game. If it failed, the Colts would get the ball in Patriots territory and have a larger probability r of then scoring a touchdown and winning the game. In the actual game, Belichick chose to go for a first down and failed -- the Colts then scored and won the game 35-34.
  • (a,10) Draw an event tree for the possible outcomes if he punted, and another event tree for the possible outcomes if he went for the first down, In each case, calculate the probability that the Patriots win the game, as a function of the probabilities p, q, and r.
  • (b,10) One estimate for the three probabilities, based on historical statistics, is p = 0.3, q = 0.6, and r = 0.5. If these were the correct probabilities, was Belichick's decision the correct one to maximize the probability that his team would win? Justify your answer.
  • (c,10) Some writers argued that these estimates of p and r were too small, because the Colts' quarterback has a long history of last-second heroics. Still assuming that q = 0.6, for what values of p and r is punting the better choice, and for what values is going for the first down the better choice? (Hint: Find an equation involving p and r that makes the two probabilities equal, then convert this into inequalities to find values for p and q that make each choice better.)

Last modified 19 November 2009