# Homework Assignment #4

#### Due on paper to me or to CMPSCI main office by 4:00 p.m., Tuesday 13 May 2008

There were originally five questions for 70 total points -- the last three-part question for 30 more points was added on 5 May 2008.

Students are responsible for understanding and following the academic honesty policies indicated on this page.

Question X.4 corrected 5 May.

• Problem X.4 (10) Suppose my prior hypothesis is that black, yellow, and chocolate Labrador retrievers are equally numerous. This is not enough information to solve the problem -- I need to have a prior that indicates the range of hypotheses I am considering. Let's say that H is the hypothesis that the three types are equally likely. The hypotheses HB, HY, and HC say that the distribution of black, yellow, and chocolate is 60-20-20, 20-60-20, and 20-20-60 respectively. My prior will then be that H is true with probability 70%, and that HB, HY, and HC have probability 10% each. I then observe three black labs and a chocolate lab. What should be my posterior estimate of the probability of each of the four hypotheses?

• Problem X.5 (10) In State M, 63% of voters are Democrats and 37% Republicans, while in State U 73% are Republicans and 27% Democrats. In State M, 90% prefer the Red Sox to the Yankees and in State U 60% prefer the Red Sox to the Yankees. Suppose I now meet a person who is either a random State M resident or a random State U resident with equal probability. What should be my estimated probability that she is from State M given that she is:
• a Democratic Red Sox fan
• a Democratic Yankee fan
• a Republican Red Sox fan
• a Republican Yankee fan

• Problem X.6 (15) We place a marker on one of the cells of a three-by-three array (a tic-tac-toe board). On each turn the marker moves one cell north, east, south, or west, with each possible move equally likely -- for example, from the northwest corner it moves south or east with probability 1/2 each. For each of three possible initial conditions (corner, side, center) determine the probability of each of the nine positions after one, two, and three turns. Determine the long-term behavior of the system and the steady-state probability if any.

• Problem X.7 (15) Repeat Problem X.6 where the marker has an equal probability of remaining where it is or moving one square -- so from the northwest corner it stays, moves south, or moves east with probability 1/3 each, and from the center it stays or moves in one of four directions with probability 1/5 each.

• Problem X.8 (20) Now consider a Markov decision process where we can decide at each turn whether to use Strategy A, which keeps the marker stationary with 1/2 probability and moves it in one of the available directions otherwise, and Plan B, which moves the marker to any of the nine squares with probability 1/9 each. The reward on any turn is 3 for the center square, 2 for a side square, and 1 for a corner square. What is the policy that maximizes the reward on the next turn? What is the policy that maximizes discounted future return with a discount factor of γ = 1/2?

• Problem X.9 (30) This problem views a single pitch of a baseball game as a two-person zero-sum game with simultaneous moves. On every pitch the state consists of a number of balls from 0 through 3 and a number of strikes from 0 through 2. On the pitch, the pitcher chooses whether to throw a good or a bad pitch, and the batter decides to swing or not swing (take). If the batter swings at a good pitch, he hits the ball in play which has a 35% chance of being a hit, so the batter gets 35 points. If the batter takes a good pitch or swings at a bad pitch the number of strikes goes up by one. If he takes a bad pitch the number of balls goes up by one. If the number of strikes reaches three the batter is out and gets no points. If the number of balls reaches four there is a walk and the batter gets a number of points to be determined below.

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• (a,5) Determine the optimal mixed strategies for each player if the count is 3-2 (three balls and two points) and a walk is worth 100 points. What is the expected return to the batter from this game?
• (b,10) Compute the expected return to the batter for the other eleven count situations. You must do this in a sensible order. For example, if the count is 2-1 a ball makes it 3-1 and a strike makes it 2-2, so to compute the value of 2-1 you must already know the values of 3-1 and 2-2.
• (c,15) Repeat parts (a) and (b) with the walk valued at 60 points rather than 100.