Java programming

Question 7 (20 points) Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. One of the basic examples of getting started with the Monte Carlo algorithm is the estimation of Pi.

Basic Idea: The idea is to simulate random (x, y) points in a 2-D plane with domain as a square of side 1 unit. Imagine a circle inside the same domain with same diameter and inscribed into the square. We then calculate the ratio of number points that lied inside the circle and total number of generated points. Refer to the image below:

We know that area of the square is 1 unit sq while that of circle is ( 1 )2 = . Now for a very large number of generated points,

24

Areacircle = Number of points generated inside cicle Areasquare Total number of points generated

i.e. = 4 Number of points generated inside cicle Total number of points generated

The beauty of this algorithm is that we dont need any graphics or simulation to display the generated points. In randomized and simulation algorithms like Monte Carlo, the more the number of iterations, the more accurate the result is. Thus, it is estimating the value of and not Calculating the value of . Implement this algorithm in Java and estimate the value of .