I've been trying to solve the problem below for over an hour doing everything but writing it out by hand. I've tried looking for patterns in strings of lengths $2, 3, 4,$ and $5$ which have probabilities $1/4, 3/8, 8/16,$ and $19/32$, respectively, of having two or more consecutive $1$s.
I'm more interested in the formula and theory behind solving the problem than the answer itself. I've searched online for permutation and probability problems without success.
Problem:
For the purposes of this puzzle, a "string" means a sequences of zeros and ones, e.g. $1011$.
Consider all the different strings of length eight — a good first step would be to figure out how many of these exist.
If you pick a length-eight string uniformly at random, what's the probability that it contains two (or more) consecutive 1s?
Problem is from the app below: https://play.google.com/store/apps/details?id=atorch.statspuzzles