Suppose that we keep throwing a fair dice n times. Also, suppose that all throws are independent and the dice is fair, i.e. 1, 2, 3, 4, 5, and 6 have equal probabilities to show up in each throw. Compute the probability that
we have the number 6 showing up in at least k throws;
we have the number 6 showing up in at least k consecutive throws.
I believed 1. was pretty easy. Simply realize the probability of not rolling a 6 in any one throw is 5/6. Then since each roll is independent so multiply this to the power of k, which gives the probability of not rolling a 6 in k throws. Then subtract that product from 1 to get the probability of rolling a 6 in at least k throws. So the answer is $1-(5/6)^k$.
I am having trouble with 2. though. The question is asking for the probability of 6 showing up in k back to back throws of the n times the die is thrown. Since each die toss is independent I feel like the process and answer for finding 2. should be the same as 1. but this is obviously not the case.
So did I do 1. wrong and how should I proceed with 2.?