- For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+\#(S)$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ and $4+\#(S) = 4+3 = 7\in S$.
Given that $S$ is randomly chosen from all $2^{15}$ subsets of $\{1,2,\ldots,15\}$ (with each subset being equally likely), what is the expected value of the number of anchors of $S$?
- Mila has four ropes. She chooses two of the eight loose ends at random (possibly from the same rope) and ties them together, leaving six loose ends. She again chooses two of these six ends at random and joins them, and so on, until there are no loose ends. At this point she has somewhere between one and four loops of rope (inclusive). Find the expected value of the number of loops Mila ends up with.
I don't really know how to start, but thanks for helping.
