I started a discussion on this general topic, that was unfortunately embedded in an earlier one about a specific question about a group modeled on English homophony. I'm sending this message to avoid a confusing half-threadjacking. I'm not sure I understood Dan's proposed probability weighting. What I think I would have done is chosen a parameter lambda between 0 and 1, and imagined a process generating random relations by starting with 1 and multiplying on the right by a randomly-chosen letter, then exiting with probability lambda, otherwise going back to stick on a new letter. If lambda is high, most of these relations will be short; if lambda is low, most will be long. We expect that it takes more relations to collapse the group if the relations are long.