I have two mathematical problems that I came up with and was unable to solve.

Problem 1)
If you roll a dice N times, what is the probability of getting a row of at least K consecutive sixes? Of course given that K < N.

Problem 2)
If you roll a dice N times, what is the average length of the maximum run of consecutive same digits? For example, if you rolled the dice a million times, and then looked up what the longest sequence of any consecutive digits was, how long would you expect that to be and what would it be on average for any number of dice rolls generally? Just to be clear here I'm talking about a consecutive sequence of any digit, not just sixes.