Stochastic Experiment and Random Event

Probability theory is a branch of mathematics that deals with the study of uncertain events or phenomena. It provides a framework for quantifying uncertainty and understanding random phenomena.
At its core, probability theory explores the likelihood or chance of various outcomes occurring in a given situation.
The key concepts of probability theory include stochastic experiments, elementary events, and random events.

Stochastic experiments

A stochastic experiment, also known as a random experiment, is an experiment or process that has a random or uncertain outcome.
Here are a two key characteristics of a stochastic experiment:

• randomness: The outcome of a stochastic experiment is determined by a random process or chance. It is not predictable with certainty;
• repetition: A stochastic experiment can be repeated multiple times in the same conditions and each repetition may result in a different outcome.

Here are some examples of stochastic experiments:

• rolling a die: The outcome of rolling a die is uncertain and can take on any of the values from 1 to 6;
• tossing a coin: The outcome of tossing a coin is uncertain and can result in either heads or tails;
• measuring the height of a randomly chosen person: The height of a randomly chosen person is uncertain and can take on any value within a certain range;
• counting the number of defects in a batch of products: The number of defects in a batch of products is uncertain and can vary from batch to batch.

Elementary events

Elementary events are the smallest indivisible results of a statistical experiment that are mutually exclusive.
For example, if we are rolling a dice, elementary events are: roll of 1, roll of 2, roll of 3, 4, 5, 6.
Each of these events is indivisible and it cannot be divided into small sub-events. In addition, in one specific stochastic experiment, only one event can occur, respectively, they are mutually exclusive. A set of all elementary events of particular experiment is called the space of elementary events.

Random events

Random event, respectively, is any result of a random experiment which can include one or more elementary events. More strictly a random event is any subset of the space of elementary events.
Random events are the main point of interest for us in solving practical problems.

In addition, we may be interested not only in one of the random events, but also in certain combinations of them.
For example:

1. Union of events A and B (A or B): we are interested in occurrence of A or in occurrence of B;
2. Intersection of events A and B (A and B): we are interested in occurrence of both A and B;
3. Complement to event A (not A): all possible random events of particular stochastic experiment excluding event A.