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Monte Carlo simulations (or Monte Carlo methods) are a class of computational algorithms that rely on repeated random sampling to obtain numerical results; the underlying idea is to use randomness to solve problems that may be deterministic in principle ^[1]. A Monte Carlo simulation constructs a stochastic model of the system under study, generates samples from the relevant probability distributions, and aggregates the sample statistics to estimate quantities such as expected values, probabilities, or integrals ^[2]. The procedure typically proceeds by defining a domain of possible inputs, generating inputs randomly from a specified probability distribution over that domain, performing a deterministic computation on each sampled input, and aggregating the results ^[3]. As the sample size increases, the law of large numbers ensures that the estimate converges to the true value, with the statistical error decreasing proportionally to the inverse square root of the number of independent samples [4]. Monte Carlo simulations are distinguished from deterministic numerical methods by their stochastic nature and from analytic probabilistic techniques by their reliance on numerical computation rather than closed-form solution ^[2]. They are applicable to high-dimensional problems and to systems whose governing equations lack tractable analytic solutions ^[3].