- Why does this method work? Isn't π about circles, not random numbers?
- The connection is through geometry and probability. The probability a random point in the square lands in the circle equals the ratio of their areas: (Area of Circle)/(Area of Square) = π/4. By sampling many points, we empirically measure this probability. This transforms a geometric area problem into a statistical counting problem, showcasing a powerful problem-solving technique.
- Why is my estimate sometimes worse when I add more points? Shouldn't it always get better?
- The estimate converges to π, but it does so statistically, not monotonically. Due to random fluctuations, adding a small batch of points can temporarily move the estimate away from π. This is normal. The Law of Large Numbers guarantees the long-term trend is toward the true value, but the path is 'noisy.' This illustrates the difference between a trend and individual observations.
- Is this a practical way to calculate π accurately?
- No, for high-precision π calculations, far more efficient deterministic algorithms exist. This method's value is pedagogical and conceptual. It demonstrates the core idea of Monte Carlo integration, which becomes immensely practical for evaluating complex, multi-dimensional integrals where deterministic methods fail, such as in financial modeling or particle physics simulations.
- What does 'uniform samples' mean?
- It means every location within the square has an equal chance of being chosen. There is no bias toward the center or edges. This is crucial for the area-ratio probability to hold. In practice, computers use pseudo-random number generators to approximate this ideal uniform distribution.