26. 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 E_i denote the event that the initial outcome is i and the player wins. The desired probability is ∑_i=2¹² P(E_i). To compute P(E_i), define the events E_in to be the event that the initial sum is i and the player wins on the nth roll. Argue that P(E_i) = ∑_n=1^∞ P(E_in).