Monte Carlo Simulation Calculator
Use repeated random sampling to estimate probabilities for coin flips, dice rolls, or π. Ideal for understanding how simulation converges to theoretical values as trial count increases.
About this calculator
Monte Carlo simulation estimates outcomes by running a large number of random trials and counting how often a target condition is met. The estimated probability is: P ≈ successes / total_simulations. For π estimation the method plots random points (x, y) in a 2×2 square; a point falls inside the unit circle when x² + y² ≤ 1. Because the circle area is π and the square area is 4, the ratio of interior points to total points multiplied by 4 approximates π. For coin flips, the simulator counts heads in 10 flips per trial. The law of large numbers guarantees that as simulations → ∞, the estimated probability converges to the true theoretical value. More simulations mean smaller variance in the estimate.
How to use
Estimate π using 10,000 simulations. Set simulation type to 'pi_estimation', num_simulations to 10000, success_criteria to 'exact', and target_parameter to 4 (the formula returns 4 for interior points, 0 otherwise). The simulation places 10,000 random points in a square; suppose 7,854 land inside the circle. Estimated π ≈ (7854 / 10000) × 4 = 3.1416 — very close to the true value of 3.14159. Increasing to 1,000,000 simulations typically yields accuracy to 3–4 decimal places.
Frequently asked questions
How many simulations do I need for an accurate Monte Carlo result?
Accuracy scales with the square root of the number of simulations — to halve the estimation error you need four times as many trials. For simple probability estimates, 10,000 simulations typically give results accurate to within 1–2 percentage points. For precise scientific or financial applications, 100,000 to 1,000,000 trials are common. The trade-off is computation time, though for browser-based calculators 10,000–50,000 simulations run near-instantly.
What real-world problems use Monte Carlo simulation?
Monte Carlo methods are used across finance (option pricing, portfolio risk estimation via Value at Risk), engineering (structural reliability, tolerances), physics (particle transport modeling), and operations research (supply chain optimization). They are especially valuable when a problem has too many variables for an analytical closed-form solution. Weather forecasting, drug trial design, and project timeline estimation all rely on Monte Carlo techniques. The common thread is replacing deterministic equations with probabilistic sampling.
Why does the Monte Carlo estimate of pi get closer to the true value with more simulations?
The π estimation works by comparing the area of a quarter-circle (π/4) to a unit square (area = 1). Each random point is an independent Bernoulli trial with success probability π/4 ≈ 0.7854. The sample proportion of successes is an unbiased estimator of this probability, and by the central limit theorem its variance equals p(1−p)/n, shrinking as n grows. With 10,000 points the standard deviation of the π estimate is roughly 0.016; with 1,000,000 points it drops to about 0.0016.