How might we approach Collatz Conjecture by probabilistic method?

Let $X_n = 1_{n \text{ is even}}$ so that $$f(n) = \frac{n}{2} X_n+\frac{3n+1}{2}(1-X_n)$$

Clearly it is not true, but you can assume the $X_n$ are independent random variables with $P(X_n=1) =P(X_n=0) =\frac{1}{2}$.

In that case for every $n$, the sequence $Y(k) = Y_n(k) = \underbrace{f \circ \ldots \circ f}_k(n)$ is bounded with probability $1$.

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Note if we look instead at $g(n) = \frac{n}{2} X_n+(3n+1)(1-X_n)$ then the sequence grows without bound almost surely... So independence of the $X_n$ is really far from the truth.

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Then you can compute the mean value and the variance of the random variable $Y_n(1),Y_n(2)$ and $Y_n(k)$ as well as $Z_K(n)=\max_{k \le K} Y_n(k)$. You'll get statement such as "with probability $p $, $\frac{Z_K(n)}{n} \le \epsilon$"