It is a good exercise to estimate π value by simulation. Let's draw a square with 2R as its side. If putting the largest circle inside the square, its radius will be R, described by the following equation:
On the other hand, the square is the product of its sides:
Dividing Equation (4) by Equation (5), we have the following result:
Reorganize it; we end up with the following equation:
In other words, the value of π will be 4* Scircle/Square. 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, 
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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...