Theory of Probability, HW #1, all sections Assignment Date: March 4, 2011; Due Date: March 11,2011 1. The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses; if the sum is either a 7 or an 11, he or she wins. If the outcome is anything else, the player continues to roll the dice until he or she rolls either the initial outcome or a 7. If the 7 comes first, the player loses; whereas if the initial outcome reoccurs before the 7, the player wins. Compute the probability of a player winning at craps. Hint: Let g1831g3036denote the event that the initial outcome is iand the player wins. The desired probability is ∑ g1842(g1831g3036)g2869g2870g3036g2880g2870.To compute g1842(g1831g3036),define the events g1831g3036,g3041to be the event that the initial sum is i and the player wins on the nth roll. Argue that g1842(g1831g3036)=∑ g1842g3435g1831g3036,g3041g3439∞g3041g2880g2869. 2. We have a stick of 9 consecutive parts. Each part is painted red or white. If none of the neighboring parts are painted white, how many different painting patterns can be done?
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