Statistical Regularity in the Monty Hall Paradox Using Proprietary Random Algorithms

Abstract

The article examines a classical problem in probability theory that has been widely discussed since the mid-1970s, known as the Monty Hall Paradox. It illustrates the discrepancy between the subjective perception of randomness and objective mathematical proofs, which are supported by appropriate computational and statistical experiments. A detailed logical analysis of the intuitive perception of the problem's solution is carried out and interpreted as another manifestation of cognitive dissonance, where even scientifically justified facts fail to change an individual's viewpoint formed within their established natural-scientific worldview. The mathematical justification of the player's optimal prize-selection strategy is presented in two probabilistic interpretations. Each of these interpretations aligns with well-known results of probability theory, which have once again been subjected to statistical verification through numerical experiments.

A distinctive feature of this study is that, in addition to the established theoretical and experimental results obtained using classical Monte Carlo methods (Mersenne Twister, PCG), quasi--Monte Carlo methods with refinements such as Sobol, Halton, Faure, and Niederreiter low-discrepancy sequences, as well as a linear congruential generator with local optimality properties, the computational procedures were also performed using previously unused random number generators. These include proprietary algorithms of a linear congruential generator in a non-improvable formulation and a quasi--Monte Carlo method based on ultra-economical Korobov lattices with small denominators $p$. In this context, the coordinate a_k/p of a lattice node of size p with denominator p is considered ``small'' within admissible limits, whereas in random algorithms there is no restriction on the length of decimal expansions relative to p.

The computations demonstrate that both of these randomness algorithms also confirm statistical regularity in convergence to the theoretical winning probabilities, both when the player keeps their initial choice and when they switch to the remaining unopened door, as the number of trials (games) increases. This result may also be interpreted inversely as confirmation of the quality of the proprietary randomness generators employed.

The article demonstrates the existence of statistical regularity in the Monty Hall game and serves as an illustrative example of the formation of probabilistic conclusions supported by statistical validation.