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.
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