Getting two consecutive $6$'s A standard six-sided die is rolled until two consecutive 6s appear. Find the expected number of rolls. Please see that this is a language problem. What do they mean by expected number of rolls? Should the answer be 36? Or 42?

Let $E$ be the expected number of rolls.   This is the mean number of roll we expect to take to witness the event.
To obtain two consecutive sixes, you must roll until you get a six followed immediately by a six (obviously).
Let the expected number of rolls until you get one six be : $F$.  Can you find what $F$ is?  (Hint: Geometric Distribution)
Then you will either get a six on the next roll (the seventh), or you don't and have to start all over from there.   There is a $5/6$ chance you don't, so we have a recursive definition for the expected number: $$E = F + 1 + \frac 5 6 E$$
