Monte Carlo Simulation Playground

About Monte Carlo Simulations

Monte Carlo methods are computational algorithms that rely on repeated random sampling to obtain numerical results. They are particularly useful for:

• Estimating probabilities: When analytical solutions are difficult or impossible
• Numerical integration: Approximating areas, volumes, and complex integrals
• Optimization problems: Finding solutions in high-dimensional spaces
• Risk assessment: Modeling uncertainty in finance, engineering, and science

Key principle: As the number of random samples increases, the empirical estimate converges to the true value (Law of Large Numbers).

Classic Monte Carlo Problems

π Estimation: Randomly place points in a square. The ratio of points inside a quarter-circle to total points approximates π/4.

Birthday Paradox: In a room of n people, what's the probability that at least two share a birthday? Surprisingly high!

Monty Hall: Should you switch doors after the host reveals a goat? Yes! Switching gives you 2/3 probability vs 1/3 for staying.

Dice & Coins: Explore probabilities of various outcomes, runs, and patterns in random sequences.