Two gamblers decided to play a game involving rolling a fair die. Player A agreed to pay Player B $10 as an "entry fee" to play. The game consists of Player A rolling a die until he rolls an 1 for the first time. An example of a possible sequence of rolls in this game is {(2, 6, 3, 5, 1)}, with the number of rolls N = 5. Assume the rolls are independent. a) Player A wins the game if it takes at least 3 rolls for the first 1 to show up. Find out the probability that player A wins one round of the game.(3 marks) b) For each round of the game, if Player A wins according to the condition in part (a), then he gets $30 from Player B. Find the expected value of Player A's profit. Is this game a good deal for Player A? Briefly explain your answer. (3 marks) c) Player A decides to play the game for 100 rounds. Find the expectation and variance of the number of games that player A wins according to the condition in part (a). (4 marks)

Question 1 Two gamblers decided to play a game involving rolling a fair diet Player A agreed to pay Player B $10 as an \"entry fee\" to play, The game consists of Player A rolling a die until he rolls an 1 for the rst time. An example of a possible sequence of rolls in this game is {(2, 6, 3, 5, 1)}, with the number of rolls N = 5, Assume the rolls are independent. a) Player A wins the game if it takes at least 3 rolls for the rst 1 to show up Find out the probability that player A wins one round of the game/3 marks) b) For each round of the game, if Player A wins according to the condition in part (a), then he gets $30 from Player B, Find the expected value of Player A's proti Is this game a good deal for Player A? Briey explain your answer. (3 marks) 0) Player A decides to play the game for 100 rounds Find the expectation and variance of the number of games that player A wins according to the condition in part (a). (4 marks )