Simulators in the study of probability
Understanding randomness begins with simple experiments: tossing a coin, rolling a die, drawing balls from a box. Such models are most often used in the initial teaching of probability theory. Open-source simulators help not only to observe the results, but also to analyse patterns in large samples.
This simplifies the learning of key concepts such as independence, probability distribution, and mathematical expectation. Among the most popular solutions are:
- R with the prob package – allows you to simulate random experiments and visualise their results;
- TinkerPlots – used in schools and colleges to build models of random processes;
- NetLogo – provides flexible tools for creating agent-based simulations, including stochastic models.
The advantage of software is that experiments can be repeated thousands of times without spending a lot of time. Users can easily change parameters, track system behaviour under different conditions, and form an intuitive understanding of probabilistic patterns.
In many countries, simulators have already become part of the curriculum. For example, GeoGebra is actively used in Italy for visual modelling of probability distributions and demonstrating dependencies between random variables. Despite its apparent simplicity, this cross-platform solution is well suited even for finding the most favourable odds from bookmaker non AAMS.
Stochastic modelling in science
Beyond the basic course, random processes are used in a wide variety of fields. Researchers use open-source tools to recreate complex systems in which uncertainty plays a decisive role. This is particularly relevant when studying dynamic and unstable environments where predictability is extremely limited.
Examples of programmes for use in science include:
- SimPy – a Python library for event-driven simulation;
- GNU Octave – an environment for numerical analysis that supports random variables;
- GillespieSSA – a package in the R language for modelling biochemical reactions using the Gillespie method;
- OpenABM – a system for agent-based modelling with support for stochastic parameters.
These tools are especially useful when direct analytical calculations are impossible. Repeated simulations allow you to identify the range of system behaviour, construct interval estimates, and test hypotheses in practice. If necessary, developers can modify the code to extend the simulator's capabilities for specific tasks.
Open-source solutions create an environment where learning is based on experimentation and interaction. Instead of mechanically memorising formulas, students work with models, evaluate results, and draw reasoned conclusions. This approach increases motivation and promotes the development of analytical thinking.
