To get a more intuitive understanding of the halting problem I want to generate some random TMs and see how they behave, what some heuristics can tell about them, etc.
The problem is that, if I correctly recall a paper I once saw, the probability for random TMs, from some reasonable distribution, to quickly halt or enter a trivial infinite loop approaches 1 as the number of states grows. That makes for boring research.
I want to know if there is a distribution of TMs in which this is not true. That is, for most TMs, it's not obvious if they halt or not. Although I'm not particular about TMs - any model of computation will do.