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Problem 1: Two cows play a game where each has one playing piece, they begin by having the two pieces on opposite vertices of an octahedron, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex originally occupied by its opponent’s piece. Because cows are not the most intelligent of creatures, they move their pieces randomly. What is the probability that the first cow to move eventually wins? Solution 1: Call the first cow to move A, and the second cow B. Let X be the probability that A will (eventually) win when it is his turn to move and the two cows are both on their original vertices. Let Y be the probability that A will (eventually) win when it is his turn to move and both cows are on one of the middle vertices. Let Z be the probability that A will (eventually) win when he is on his original vertex and B is on one of the middle vertices. And let W be the probability that that A will (eventually) win when he is on one of the middle vertices and B is on his original vertex. From these designations we get: X = Y, because from their original vertices they must each transition to one of the middle vertices Y = ¼ + 1/16 X + ¼ Y + 1/8 Z + 1/8 W Because A can either win immediately with probability 1/4, or both cows return to their original vertices with probability 1/16, or both stay on the middle vertices with probability ¼, or have A return to his original vertex and B stay in the middle with probability 1/8, or B return to his original vertex and A stay in the middle with probability 1/8 Z=½Y+¼W Because A must return to one of the middle vertices while B could either stay in the middle, probability ½, or return to his original vertex, probability ¼. W=¼+½Y+¼Z Because, from one of the middle vertices, A can either win immediately with probability ¼, or stay on one of the middle vertices with probability ½, or return to his original vertex with probability 1/4. This gives us a system of equations and X is the value we seek. Solving gives us X = 14/25, Y = 14/25, Z = 11/25, and W = 16/25. Problem 2: A fair coin is tossed repeatedly. Find the probability of obtaining five consecutive heads before two consecutive tails. Solution: Let be the desired probability, be probability given first flip is heads, be probability given first flip is tails. by adding cases where first flip is heads or tails. because either the second flip is tails (failure) or heads (begins a new run with heads). So . Now. if the first flip is heads, then either: - Four heads follow (success) with probability . - Up to three heads follow, but then tails. In order to have a chance at success, the flip after tails must be heads. If it is, then a new run begins with heads, which is neutral. (That means success is independent of these flips.) The combined probability of TH, HTH, HHTH, HHHTH is . Then Multiplying by . , . Practice Problems 1. Two cowboys participate in a duel. They take turns shooting at one another until one of them is hit. Cowboy 1 has an accuracy of 1/4 and cowboy 2 has an accuracy of 1/3. What is the probability that cowboy 1 will win the duel? 2. Two cows play a game where each has one playing piece. They begin by having the two pieces on opposite vertices of a square, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex originally occupied by its opponent’s piece. Assuming the cows move their pieces randomly, what is the probability that the first cow to move eventually wins? 3. Again two cows play a game where each has one playing piece. They begin by having the two pieces on opposite vertices of a hexagon, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex originally occupied by its opponent’s piece. Assuming the cows move their pieces randomly, what is the probability that the first cow to move eventually wins? 4. Erica, Nate, and Jason play a game in which they take turns flipping a fair coin. Erica flips first, followed by Nate, followed by Jason, and so on. The first person whose coin flip matches their predecessors flip wins the game. What is the probability that Nate will win? 5. You throw a coin multiple times. What's the average amount of throws required to obtain 2 heads in a row? 6. Two players, A and B, start with 8 and 4 coins, respectively. They flip a fair coin, and based on the results of each flip, one player pays the other one coin. The game ends when one player goes broke. What is each player's probability of winning, and how long, on average, does the game last? 7. Suppose you are on a number line, and you start at 1. You repeatedly roll a die. If it lands on 1 or 2, go down one integer. If it isn't a 1 or a 2, go up one integer. What is the probability that you land at zero at some point? 8. Find the probability that if a fair coin is tossed infinitely and the results are recorded in a sequence, "HH" will occur before "THTH" Answers: 1. 1/2 2. 2/3 3. 4/7 4. 4/7 5. 4 6. Player A = 2/3, Player B = 1/3, Expected length of game = ??? 7. ½ 8. 3/4