Monte Carlo Simulation

Brief introduction to the Monte Carlo method: Monte Carlo (Monte Carlo) is a place name located in Monaco, the gambling capital, symbolizing probability. The Monte Carlo method was proposed by the famous mathematician John von Neumann and was born during the "Manhattan Project" in the United States in the 1940s.

The principle is to understand a system through a large number of random samples, thereby obtaining the value to be calculated.

The principle of estimating pi**

A square with an inscribed circle: the ratio of the circle's area to the square's area is π/4 (you can derive this yourself from the area relationship).

Inside this square, randomly generate n points (these points follow a uniform distribution) and calculate whether their distance from the center point is greater than the circle's radius to determine whether they fall inside the circle. Count the number of points inside the circle, and the ratio to n multiplied by 4 is the value of π. In theory, the larger n is, the more accurate the calculated value of π.

We randomly generate N randomly distributed points and determine whether they are inside the circle, then estimate the approximate value of pi using the formula.

This tool can be used to demonstrate probability experiments—as long as the generated random points are sufficiently uniform and random, and there are enough random points, the final estimation result becomes more precise.

Of course, the Monte Carlo method can also be used for many other purposes in various fields; estimating the value of π is just one very interesting small application scenario.