Consider the condensed collatz conjecture
if $x$ odd then $f(x)=(3x+1)/2$: if $x$ even $f(x) = x/2$: Continue until $x = 1$ or find an $x$ in the natural numbers that will not hit $1$.
The equation is condensed in this way because originally every $3x+1$ step would be trivially immediately be followed by a $x/2$ step under iteration.
My question is what would go into a proof or disproof that on average for a natural number $x$ each step in the iteration is equally likely I.e. each step has $50\%$ probability in the long run?
Personally, I've never liked the condensed collatz function as you give it because if $x$ is odd, then two calculations are performed, but if $x$ even, then only one calculation is done. One cannot really count "steps", i.e., how many operations it takes to reduce to 1.
That aside, to answer your question: note that $x<\frac{3x+1}{2}$, so if each step were equally likely, $n$ would increase and never reduce to 1. It seems to me, that for any number (especially large ones), there have to be more steps with $n$ even in order for the sequence to stop.