Suppose you have one undecidable statement, U, out of s possible statements. A random program of length n contains U at least once with probability 1-(1-1/s)^n. If the program is of infinite length, it contains U with probability 1. So I would have thought the chances of undecidability increase as the program gets longer.
Clearly my intuition is wrong, per the paper, but I don't think the result is immediately obvious.
So if you make a random TM, it's almost guaranteed that it'll get eventually "trapped" in one of these easily decidable infinite loops.
There isn't really a single undecidable statement, it would be a complex subroutine. My intuition would be its much more likely a halt or indef loop (both relatively simple) would appear before program control would transfer to the undecidable part, in most cases.
I.e. if something undecidable requires 60 symbols in a row in correct order, but halting requires one, than probability of a halt at any given spot seems much higher on average.