(1) events can be repeated under basically the same conditions, such as shooting the same target with the same gun many times. There is only a single accidental process and its repeatability cannot be judged, which is not a random event.

(2) Under basically the same conditions, an event may be manifested in many ways, and it is uncertain in advance which specific way it will happen. For example, no matter how to control the firing conditions of guns, it is impossible to accurately predict the location of the impact point before firing. The only possible process is not a random event.

(3) All the possibilities of an event appearing in various ways can be predicted in advance, and so can the probability of it appearing in some way, that is, the frequency in the process of repetition. For example, when a large number of shells are fired, the impact points are normally distributed, and each impact point has a certain probability in a certain range. A phenomenon with no definite probability when it occurs repeatedly is not a random event of the same process.

Assume that there are inevitable events and impossible events in the real world, and random events are phenomena and processes between inevitable events and impossible events. Specific events in the fields of nature, society and thinking are random.

The inevitable and certain events in the macro world will have random deviations in details. The motion state of a single object in the microscopic world is random. Whether the products are qualified in material production, the price fluctuation of commodities, the appearance of errors in scientific experiments and the interference in information transmission are often random.

The study of random events, random variables, random sampling and random functions is an important part of probability theory and mathematical statistics in modern mathematics, and it is widely used in natural science, social science and engineering technology.

For a random event, we can discuss its possible probability and reflect the possibility of the event. A large number of repeated random events show statistical regularity. Statistical law is the overall law of a large number of random phenomena, which dominates the state of stochastic systems.