CMPSCI 291b: Reasoning About Uncertainty

Final Exam

David Mix Barrington

15 May 2008

Directions:

• Answer the problems on the exam pages.
• There are ten problems for 120 total points. Actual scale was A=110, C=75.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first six questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• The other questions have numerical answers -- 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.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 10 points
  Q7: 10 points
  Q8: 20 points
  Q9: 30 points

  Total: 120 points

• Question 1 (10): True or false with justification: Consider a version of Two-Finger Morra where each player decides to put out one or two fingers, but this time player E wins the product of the two numbers if this product is even and loses the product if the product is odd. Then if both play optimally, E will win 2 points.

• Question 2 (10): True or false with justification: It is estimated that a British airman in World War II had a 5% chance of dying on any given bombing mission. If we assume that this was the exact probability, and that the events of dying on different missions were independent, then an airman was certain to die at some point during his first 20 missions.

• Question 3 (10): True or false with justification: If a nine-digit Social Security number is uniformly chosen from all possible nine-digit numbers, then the probability that the number has four or more 0's is less than (9 choose 4) times 10^-4.

• Question 4 (10): True or false with justification: If a nine-digit Social Security number is chosen as in Question 3, then the chance of the number having exactly one 0 is exactly (0.1)(0.9)⁸.

• Question 5 (10): True or false with justification: There are more than 1250 five-card poker hands that contain the ace, 2, and 3 of hearts. (Here we consider the same cards in a different order to be the same hand.)

• Question 6 (10): True or false with justification: Of the five-card poker hands counted in Question 5, exactly 45 are flushes or straight flushes.

• Question 7 (10): My experience is that 75% of the dogs in my area are friendly. Friendly dogs growl at me 10% of the time, and unfriendly dogs growl at me 70% of the time. If I meet a previously unknown dog who growls at me, what should be my estimate of the probability that it is friendly? What should be my estimate of the probability that it is friendly if it does not growl at me?

• Question 8 (20): On a birdwatching trip, I observe a blue parrot. Consulting my bird book, I find that 10% of the blue parrots in this area are Norwegian Blues, and the other 90% are Common Blues. It also says that there are four distinguishing characateristics of the Norwegian Blue (lovely plumage, rests on back, pines for fjords, stuns easily) -- each characteristic is observed 60% of the time in Norwegian Blues but only 20% of the time in Commmon Blues, and the different characteristics are independent of each other.

  If I observe exactly three of the four Norwegian Blue characteristics in my parrot, what is my new estimate of the probability that it is a Norwegian Blue?

• Question 9 (30): This question concerns a Markov decision process with three states, called A, B, and C, and two possible actions S (stay) and J (jump). On action S, the state always remains the same on the next step. On action J, it moves to one of the other two states with probability 0.5 each. The reward function is 3 for state A, 2 for state B, and 0 for state C.

  • (a,10) What policy (action to be taken in each state) maximizes the expected return on the next move? What is the expected return from each state?

  • (b,10) Consider the policy that takes S from state A and takes J from state B or C. What is the matrix of the Markov chain resulting from this policy? What is the long-term behavior of this Markov chain? Justify your answre.

  • (c,10) What policy on the first turn maximizes the total expected reward on the first two turns? (There is no discounting here.) (You should assume that on the second turn, you will apply the policy found in part (a).)

Last modified 20 May 2008