Given a graph $G$, I would like to determine a method for randomly generating subgraphs $G'$ with the following properties:

Each edge of $G$ has at least some probability $p$ of going into $G'$

The tree-width of $G'$ is bounded.

I would like to find a method of generating $G'$ so that $p$ is as large as possible. Note that any $p = 1/n$ is possible. if you select a random spanning tree. Are there larger values of $p$ which are possible to achieve?