Theoretical probability and empirical frequency
Supplemental study note
Original supplemental explanation aligned to the probability learning model
A theoretical probability describes the mathematical model. An empirical frequency is what happened in a finite experiment.
If an event has theoretical probability p, a long seeded simulation may produce a cumulative success ratio close to p, but the experiment does not prove the probability. The exact model and the random experiment answer different questions.
The simulation lab deliberately exposes its seed so the same experiment can be reproduced, discussed, and debugged.