Question

Write a Java program that uses the Monte Carlo method to estimate the value of PI.

This method uses the unit circle inscribed in a square with sides of length 2 and random numbers to perform the estimation.

The estimation works as follows:

• Two random numbers are generated during each iteration of a loop.

• The random numbers are each in the range of -1 to 1. One random number is the x-coordinate and the other is the y-coordinate. Therefore the point generated by these two coordinates lies inside the square.

• Some of the points generated will lie on or inside the circle. This will be true if the following condition holds:

sqrt(x2 + y2) <= 1.

• PI can be estimated by looking at the ratio of the points generated that lie on or inside the circle to the total number of points generated (some of which will lie outside the circle).

• The law of large numbers suggests that as the number of iterations (i.e number of points generated) becomes large, the above ratio will approach the ratio of the area of the circle to the area of the square, that is:

As number of trials approaches ∞,

(points on or inside circle / total points generated) approaches (area of circle)/(area of square).

• Since the area of the unit circle is PI units and the area of the square is 4 units Pi can be estimated as follows:

(points on or inside circle)/(total points) = (area of circle)/(area of square)

​​​​​= PI / 4

so that,

​​PI = 4 * (points on or inside circle)/total points.

Your program should do the following:

• Ask the user to input a positive integer for the number of trials to be used for the estimation.

• Estimate PI using the Monte Carlo method. Note: You MUST use Math.random() to generate your random values.

• Output the estimate of PI.

• Ask the user if they would like to try another estimation.

Answer 1

thanks for the question, here is the completed code in Java with inline comments, the larger the value of SIZE the more close the value of PI will be calculated, here is the code

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import java.util.Random;



public class MonteCarlo {



    public static void main(String[] args) {

        Random random = new Random();

        // running the problem for 1Million times

        int SIZE = 1000000;

        double countWithinCircle = 0;



        for (int i = 1; i <= SIZE; i++) {

            // generating a random number between -1 and 1

            double x = -1 + random.nextDouble()*2;

            double y = -1 + random.nextDouble()*2;

            // checking if the distance is less than equal to the radius of the unit circle

            // if yes we count it as within the circle

            if (Math.sqrt(x * x + y * y) <= 1) {

                countWithinCircle += 1;

            }



        }

        // to calculate the value of PI we need to multiply the ratio with 4

        System.out.println("Value of PI as calculated: "+ 4*countWithinCircle/SIZE);





    }

}

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