METHOD OF USING LAPLACE'S INTEGRAL THEOREM TO FIND THE PROBABILITY OF DIFFERENCE FROM THE CONSTANT PROBABILITY OF RELATIVE FREQUENCY IN INDEPENDENT TRIALS

Modern big-data tasks require methods for evaluating statistical deviations. This study proposes a method to estimate the probability of relative frequency deviation from a constant in independent trials. Using Laplace's theorem, this approach overcomes the large sample limitations of Bernoulli’s formula. The methodology analyzes the distribution asymptotic behavior and applies a Laplace function to calculate the boundary conditions. Experiments with a 400-element sample showed that, with an event probability of 0.1, a 0.03 deviation was achieved with 0.9544 probability. Increasing data volume reduces estimated variance while maintaining prediction accuracy. This method enhances statistical modeling for quality control and risk analysis.