Probability Simulators

Mathematical Principles & Theorems

Monte Carlo simulation empirically demonstrates fundamental probability theorems: The Weak Law of Large Numbers \(\lim_{n \to \infty} P(|\bar{X}_n - \mu| \ge \epsilon) = 0\), the Central Limit Theorem stating sums of i.i.d. random variables converge to Gaussian \(\mathcal{N}(n\mu, n\sigma^2)\), Binomial distribution \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\), and the Birthday Collision Theorem \(P(\ge 1 \text{ match}) = 1 - \prod_{i=1}^{n-1}(1 - i/365)\).

Operating Instructions

• Select an experiment: Coin Toss, Dice Roll, Monty Hall Problem, or Birthday Paradox.
• Configure trial count \(N\) or group size using the numeric input.
• Click Run / Flip / Roll to execute Monte Carlo batches.
• Observe live distribution histograms and empirical convergence toward theoretical expectation lines.