I think that your question is one of many answers and interpretations, making it a good question, but very hard to answer. Thus, my response will be very vague and simple, yet I am willing to expand if needed.
Classical mechanics has the advantage of being deterministic, Statistical Mechanics (although it is in theory classical), and Quantum Mechanics are not deterministic, they are statistical, as you have noted. Hence, they employ several statistical methods. With that been said, some are more efficient than others, thus, physics uses this methods in order to come up with methods (namely, equations) of its own to solve particular problems.
For example the Montecarlo Method would not be very productive when studying the behavior of several particles on a closed space (imagine running so many computations!), on the other hand the Maxwell-Boltzmann Distribution, which is derived from a series of analytical statistical methods, is very productive. Another example of how physics applies statistics to make its own methods, perhaps a more commonly known one, is the fact that the probability amplitude of finding a particle described by $\Psi (x, t)$ on time $t$, is $\Psi*\Psi$, a concept widely used in Quantum Mechanics and particle physics.
In conclusion, physics does not use only one statistical method, it analyzes several methods to come up with methods of its own. Also, the "type" of statistics used depends upon the field, particularly upon how deterministic the area is. Thus, when speaking QM one speaks in an statistical way, but when speaking about Classical Mechanics (a pendulum's period, for example :p) one speaks in a more deterministic way, hence employing little, if any, statistical methods.
Let me know if you have any other questions, I hope this is of some help to you!
