Abstract This article discusses the probability of the binomial distribution. Many problems of this topic lead students to solve them in one way. However, a deep study of this topic from various sources leads to solving them using the following formulas. All these situations are united by a common approach to content, but have different approximation methods. These basic methods not only simplify complex probability problems, but also improve understanding for both students and researchers. Step-by-step examples illustrate the transition from the binomial to the Poisson distribution, as well as the application of the Moivre-Laplace theorem, providing clarity and ease of understanding. The purpose of this article is to illuminate these important concepts in a way that is accessible and interesting to all readers. In our days when AI can solve any problems this article will be useful, because it gives a look at these problems from above. And this approach will allow those who wants to understand such problems and systematize their solutions.