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Question 2: Alexa┬╣, Tri2, and Zoltan play the OppPlayER game: In one round, each player flips a fair coin. 1. Assume that not all flips are equal. Then the coin flips of exactly two players are equal. The player whose coin flip is different is called the odd player. In this case the odd player wins the game. For example, if Alexa flips tails, Tri flips heads, and Zoltan flips tails then Tri is the odd player and wins the game. 2. If all three coin flips are equal, then the game is repeated. Below. this game is presented in pseudocode: 1your friendly TA Panother friendly TA Syet another friendly TA Algorithm OpdPLAYER: // all coin flips made are mutually independent each player flips fair coin: if not all coin flips are equal then the game terminates and the odd player wins else OpdPlayER endif What is the sample space? Define the event A "Alexa wins the game" Express this event as subset of the sample space Use your expression from the previous part to determine Pr(A). Use symmetry to determine Pr(A). Explain your answer in plain English and few sentences. Hint: What is the probability that Tri wins the game? What is the probability that Zoltan wins the game? Question 3: Consider the set S [2,3,5,30} We choose uniformly random element T from this set. Define the random variables 1 divisible by 2. x 0 otherwise if x is divisible by 3. Y 0 otherwise. divisible by 5. Z 0 otherwise Is the sequence X.Y Z of random variables pairwise independent? As always justify your answer Is the sequence X. Z of random variables mutually independent? As always, justify your answer Question 4: Let and be realnumbers. You flip fair and independent coin three times. For 1.2.3 let if the i-th coin flip results in heads, fa if the i-th coin flip results in tails. Define the random variables x For each of the following questions, justify your answer Assume that =b. Are the random variables x and Y independent? Assume that =0 a. Are the random variables X and y independent? Assume that to and b== Are the random variables X and y independent? Assume that #0,6=0,ath and b # -a. Are the random variables x and y independent?

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