for a big odd number (n) the next number could be

odd, 1/2 chance (3n+1)/2 which is near average 3/2 n

2*odd, 1/4 chance (3n+1)/4 which is near average 3/4 n

=3/4 chance sum = near average  =(3/2/2+3/4/4)*(4/3) =5/4 n

4*odd 1/8 chance (3n+1)/8 which is near average 3/8 n

=7/8 chance sum = near average =(3/2/2+3/4/4+3/8/8)*(8/7) = 9/8 n

expand fully n leads to n on average

but if it hits lower numbers it will lead to 1

unless it can increase forever

This looks like an informal argument for the truth of the Collatz conjecture. A fun exercise if you have the time is to attempt a formal proof using mathematical induction. It's frustrating how provable the conjecture looks, and how impossible producing an actual proof seems…after a few hours of trying :)