Game 1: You are playing a game where you get n turns. Each of your turns...

 

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Game 1: You are playing a game where you get n turns. Each of your turns involves flipping a coin a number of times. On the first turn, you have 1 flip, on the second turn you have two flips, and so on until your n-th turn when you flip the coin n times. All the flips are mutually independent. The coin you are using is biased to flip Heads with probability p. You win a turn if you flip all Heads in that turn. Let X be the r.v. equal to the number of winning turns. Calculate E[X] and Var[X]. [10 marks]. Game 2: You are playing a game where you get n turns. Each of your turns involves flipping a coin a number of times. On the first turn, you have 1 flip, on the second turn you have two flips, and so on until your nth turn when you flip the coin n times. All the flips are mutually independent. The coin you are using is biased to flip Heads with probability p. You win a turn if you flip at least one Heads in that turn. Let Y be the r.v. equal to the number of winning turns. Calculate E[Y] and Var[Y]. [15 marks]. Game 1: You are playing a game where you get n turns. Each of your turns involves flipping a coin a number of times. On the first turn, you have 1 flip, on the second turn you have two flips, and so on until your n-th turn when you flip the coin n times. All the flips are mutually independent. The coin you are using is biased to flip Heads with probability p. You win a turn if you flip all Heads in that turn. Let X be the r.v. equal to the number of winning turns. Calculate E[X] and Var[X]. [10 marks]. Game 2: You are playing a game where you get n turns. Each of your turns involves flipping a coin a number of times. On the first turn, you have 1 flip, on the second turn you have two flips, and so on until your nth turn when you flip the coin n times. All the flips are mutually independent. The coin you are using is biased to flip Heads with probability p. You win a turn if you flip at least one Heads in that turn. Let Y be the r.v. equal to the number of winning turns. Calculate E[Y] and Var[Y]. [15 marks].