Estimating π by Throwing Random Darts
The Monte Carlo method approximates mathematical values through random sampling. To estimate π, random points are scattered inside a unit square (1×1). Any point where x²+y²≤1 falls inside the quarter-circle. Since the area of the quarter-circle is π/4, the ratio of inside points to total points converges to π/4, giving the estimate π ≈ 4 × inside / total. This visual approach makes an abstract statistical concept tangible. The Monte Carlo method is used in finance, physics, AI, and many other fields where exact solutions are impractical.
Frequently Asked Questions
Points are placed randomly each time, so the estimate varies. With more points the variance decreases, but each individual run still differs slightly.
Monte Carlo convergence is proportional to 1/√n. To reduce error by 10×, you need 100× more points. Practical precision has limits, but the method is powerful for hard-to-solve problems.
Blue dots landed inside the quarter-circle (x²+y²≤1). Red dots landed outside. The ratio of blue to all dots, multiplied by 4, gives the π estimate.