If a fair coin is flipped $n$ times, what are the expected values (in terms of $n$) of: 1. The length of the longest run? 2. The number of runs of length 1? 3. The total number of runs?
I tried working it out for some small values, but could not find a sufficiently exploitable/noticeable pattern to solve any of these.
For example, for 1 coin the answers are 1,1,1. For 2 coins, the answers are 1.5,1,1.5. For 3 coins, the answers are 2,10/8,2. So for questions 1 and 3, I suspect the answer is $\frac n 2 +0.5$, but I don’t know how to prove it. For question 2, I don’t know how to prove it AND I haven’t found a pattern.