Define
a_n = probability that the 1's counter is 0 after the nth roll
b_n = probability that the 1's counter is 1 after the nth roll
c_n = probability that the game ends after the nth roll
you know:
a_1 = 5/6, b_1 = 1/6, c_1 = 0
a_{n+1} = 5/6 a_n + 1/6 b_n
Find similar terms for b_{n+1} and c_{n+1}, then solve the recurrence, and find the average of the c's to answer your first question.
