I have two fair dice. On each die, each of the numbers (1,2,3,4,5,6) comes up with...

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I have two fair dice. On each die, each of the numbers (1,2,3,4,5,6) comes up with equal probability. We play the following dice game: You start with 0 points. In each round, I roll two dice. If the number 6 appears on either die (or on both dice), the game is immediately over and your payoff is 0 dollars. Otherwise, you add to your current score the two numbers showing on the dice. For example, if the first roll is (5,3), then your total score is 5+3 = 8. You now decide whether to keep playing. If you choose to stop playing, the game is over and your payoff is your current score in dollars. If you choose to keep playing, then I roll the two dice again. I keep rolling two dice until you choose to stop playing, or I roll at least one 6, in which case you lose everything. For example, suppose the first three rolls are (5,3), (2,2), and (1,5). If you choose to stop playing, the game is over and your payoff is 5+3+2+2+1+5 = 18 dollars. However, if you choose to keep playing and the next roll is (6,4), then the game is over and your payoff is 0 dollars. (a) Suppose you have a score of N, and decide to keep playing. I throw the two dice. Clearly explain 11 why there is a probability that you lose everything. 36 (b) Suppose you have a score of N, and decide to keep playing. I throw the two dice. Clearly explain why the expected value for the increase in your score is -11N + 150 36 (c) Determine the optimal strategy for this game. Clearly and carefully justify your answer. I have two fair dice. On each die, each of the numbers (1,2,3,4,5,6) comes up with equal probability. We play the following dice game: You start with 0 points. In each round, I roll two dice. If the number 6 appears on either die (or on both dice), the game is immediately over and your payoff is 0 dollars. Otherwise, you add to your current score the two numbers showing on the dice. For example, if the first roll is (5,3), then your total score is 5+3 = 8. You now decide whether to keep playing. If you choose to stop playing, the game is over and your payoff is your current score in dollars. If you choose to keep playing, then I roll the two dice again. I keep rolling two dice until you choose to stop playing, or I roll at least one 6, in which case you lose everything. For example, suppose the first three rolls are (5,3), (2,2), and (1,5). If you choose to stop playing, the game is over and your payoff is 5+3+2+2+1+5 = 18 dollars. However, if you choose to keep playing and the next roll is (6,4), then the game is over and your payoff is 0 dollars. (a) Suppose you have a score of N, and decide to keep playing. I throw the two dice. Clearly explain 11 why there is a probability that you lose everything. 36 (b) Suppose you have a score of N, and decide to keep playing. I throw the two dice. Clearly explain why the expected value for the increase in your score is -11N + 150 36 (c) Determine the optimal strategy for this game. Clearly and carefully justify your answer.

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a Suppose you have a score of N and decide to keep playing I throw the two dice Clearly explain why there is a probability that you lose everything Th... View the full answer