Linked List Random Node - Insight

As you traverse, keep a current selection. For the $i$-th node, replace your selection with probability $1/i$. After seeing all $n$ nodes, each has probability $1/n$ of being selected.

Why does this work? Node 1 is kept with probability $1 \times (1 - 1/2) \times (1 - 1/3) \times ... \times (1 - 1/n) = 1/n$. The same holds for all other nodes by similar calculation.