It is a good exercise to estimate pi by the Monte Carlo simulation. Let's draw a square with 2R as its side. If we put the largest circle inside the square, its radius will be R. In other words, the areas for those two shapes have the following equations:

By dividing equation (4) by equation (5), we have the following result:

In other words, the value of pi will be 4* Scircle/Ssquare. When running the simulation, we generate n pairs of x and y from a uniform distribution with a range of zero and 0.5. Then we estimate a distance that is the square root of the summation of the squared x and y, that is,. Obviously, when d is less than 0.5 (value of R), it will fall into the circle. We can imagine throwing a dart that falls into the circle. The value of the pi will take the following form:

The following graph illustrates these random points within a circle and within a square:

The Python program to estimate the value of pi is presented as follows:

import...