How might we approach Collatz Conjecture by probabilistic method - or similar?
I was thinking along the following lines. Suppose we select some integer at random. What is the expected value of the number that it converges to?
Am I right in thinking if so much as a single number converges to infinity, this would set the answer to infinity?
Therefore if we can prove the expected value is finite we have proven no sequence ascends infinitely. If we can prove the expected value is $>1$ there are nontrivial loops and if we can prove the expected value is $1$ we prove the conjecture in full.
We would have to define the lowest integer in some loop as the convergent point, or find some other general definition of the value of a loop.