If my calculations are right the probability that the next number in Collatz'sequence has increased its bitlength is equal to $31.25 \%$
Bitlength meaning; the length from a fixed position of least significant bit equal to either $0$ or $1$ to the most significant bit that has only the value $1$.
The results are done on the shortcut definition of the Collatz map.
I came up with this probability based on neighbouring bitpatterns. The first based on three pluss a past state yielding: $12.5\%$ for $\frac{3n+1}{2}$ if odd, and I included the probability $50\%$ for $\frac{n}{2}$ if even.
Are there any literature on the probabilities of the bitlength of numbers in Collatz sequence of the function $\frac{3n+1}{2} \text{ if } n \equiv 1 \mod 2$ and $\frac{n}{2} \text{ if } n \equiv 0 \mod 2$ ?
I would like to know if the percentage has been found anywhere else and/or is written in the literature so I can study what has been found. Or if someone could actually test this based on large amounts of initial numbers say up to $1000000$?
If the percentages are correct it also suggest heuristically that there is an orbit trending towards $1$.
Just an image of the distribution over 200 experiments of your probability-value $ ~31.25% $ (my overall result for 200 experiments is $33.338$)
where one experiment means:
$1\,000\, 000$ random numbers $ x_1 \lt 2^{31}$ are tested, from where the probability of increase by one or more bits $\text{binary_length}(x_2) \gt \text{binary_length}(x_1)$ is documented. See the image below
(data generated in about one minute using Pari/GP)
The following picture is for the $3n+1; n/2$ version.
Data