"The following heuristic probabilistic argument support the 3x + 1 (Collatz) Conjecture ... Pick an odd integer [tex]n_{0 }[/tex] at random and iterate the

function T until another odd integer [tex]n_{1}[/tex] occurs. Then 1/2 of the time, [tex]n_{1} = \frac{3(n_{0}+1)}{2}[/tex], 1/4 of the time [tex]n_{1} = \frac{3(n_{0}+1)}{4}[/tex], 1/8 of the time [tex]n_{1} = \frac{3(n_{0}+1)}{8}[/tex] , and so on. If one supposes that the function T is sufficiently 'mixing'' that successive odd integers in the trajectory of n behave as though they were drawn at random (mod [tex]2^{k}[/tex]) from the set of odd integers (mod [tex]2^{k}[/tex]) for all k, then the expected growth in size between two consecutive odd integers in such a trajectory is the multiplicative factor,

[tex]( \frac{3}{2} )^{\frac{1}{2}} * ( \frac{3}{4} )^{\frac{1}{4}} * ( \frac{3}{8} ) ^{\frac{1}{8}} *[/tex] ... = [tex]\frac{3}{4} < 1[/tex].

Consequently, ... "

Source Links: http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node3.html;

'Collatz (3x+ 1) Con jecture',

https://en.wikipedia.org/wiki/Collatz_conjecture.

Does the above argument makes sense? Is it correct?

If that argument is correct, then it suggests a counterexample to Collatz Conjecture may exist. Do you agree?