# Estimating the value of Pi using Monte Carlo

Monte Carlo estimation
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
Estimation of Pi
The idea is to simulate random (x, y) points in a 2-D plane with domain as a square of side 2r units centered on (0,0). Imagine a circle inside the same domain with same radius r 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:

Random points are generated only few of which lie outside the imaginary circle

We know that area of the square is  unit sq while that of circle is .  The ratio of these two areas is as follows :

Now for a very large number of generated points,

that is,

The beauty of this algorithm is that we don’t need any graphics or simulation to display the generated points. We simply generate random (x, y) pairs and then check if . If yes, we increment the number of points that appears inside the circle. In randomized and simulation algorithms like Monte Carlo, the more the number of iterations, the more accurate the result is. Thus, the title is “Estimating the value of Pi” and not “Calculating the value of Pi”. Below is the algorithm for the method:
The Algorithm
1. Initialize circle_points, square_points and interval to 0.
2. Generate random point x.
3. Generate random point y.
4. Calculate d = x*x + y*y.
5. If d <= 1, increment circle_points.
6. Increment square_points.
7. Increment interval.
8. If increment < NO_OF_ITERATIONS, repeat from 2.
9. Calculate pi = 4*(circle_points/square_points).
10. Terminate.
The code doesn’t wait for any input via stdin as the macro INTERVAL could be changed as per the required number of iterations. Number of iterations are the square of INTERVAL. Also, I’ve paused the screen for first 10 iterations with getch() and outputs are displayed for every iteration with format given below. You can change or delete them as per requirement.

x y circle_points square_points - pi

Examples:

INTERVAL = 5
Output : Final Estimation of Pi = 2.56

INTERVAL = 10
Output : Final Estimation of Pi = 3.24

INTERVAL = 100
Output : Final Estimation of Pi = 3.0916

## C++

 /* C++ program for estimation of Pi using Monte   Carlo Simulation */#include  // Defines precision for x and y values. More the// interval, more the number of significant digits#define INTERVAL 10000using namespace std; int main(){    int interval, i;    double rand_x, rand_y, origin_dist, pi;    int circle_points = 0, square_points = 0;     // Initializing rand()    srand(time(NULL));     // Total Random numbers generated = possible x    // values * possible y values    for (i = 0; i < (INTERVAL * INTERVAL); i++) {         // Randomly generated x and y values        rand_x = double(rand() % (INTERVAL + 1)) / INTERVAL;        rand_y = double(rand() % (INTERVAL + 1)) / INTERVAL;         // Distance between (x, y) from the origin        origin_dist = rand_x * rand_x + rand_y * rand_y;         // Checking if (x, y) lies inside the define        // circle with R=1        if (origin_dist <= 1)            circle_points++;         // Total number of points generated        square_points++;         // estimated pi after this iteration        pi = double(4 * circle_points) / square_points;         // For visual understanding (Optional)        cout << rand_x << " " << rand_y << " "             << circle_points << " " << square_points             << " - " << pi << endl             << endl;         // Pausing estimation for first 10 values (Optional)        if (i < 20)            getchar();    }     // Final Estimated Value    cout << "\nFinal Estimation of Pi = " << pi;     return 0;}

## Java

 // Java program for estimation of Pi using Monte//Carlo Simulationimport java.util.*;import java.io.*;import java.util.concurrent.ThreadLocalRandom; class GFG{    // Defines precision for x and y values. More the    // interval, more the number of significant digits    static int INTERVAL = 10000;         // Driver code    public static void main(String[] args)throws IOException    {        double rand_x, rand_y, origin_dist, pi=0;        int circle_points = 0, square_points = 0;                 // Total Random numbers generated = possible x        // values * possible y values        for (int i = 0; i < (INTERVAL * INTERVAL); i++) {                  // Randomly generated x and y values in the range [-1,1]            rand_x = Math.random()*2-1;            rand_y = Math.random()*2-1;                  // Distance between (x, y) from the origin            origin_dist = rand_x * rand_x + rand_y * rand_y;                  // Checking if (x, y) lies inside the define            // circle with R=1            if (origin_dist <= 1)                circle_points++;                  // Total number of points generated            square_points++;                  // estimated pi after this iteration            pi = ((4.0 * circle_points) / square_points);                  // For visual understanding (Optional)            //System.out.println(rand_x+" "+rand_y+" "+circle_points+" "+square_points+" - "+pi);        }              // Final Estimated Value        System.out.println("Final Estimation of Pi = " + pi);    }} // This code is contributed by shruti456rawal

## Python

 import random INTERVAL = 1000 circle_points = 0square_points = 0 # Total Random numbers generated= possible x# values* possible y valuesfor i in range(INTERVAL**2):     # Randomly generated x and y values from a    # uniform distribution    # Range of x and y values is -1 to 1    rand_x = random.uniform(-1, 1)    rand_y = random.uniform(-1, 1)     # Distance between (x, y) from the origin    origin_dist = rand_x**2 + rand_y**2     # Checking if (x, y) lies inside the circle    if origin_dist <= 1:        circle_points += 1     square_points += 1     # Estimating value of pi,    # pi= 4*(no. of points generated inside the    # circle)/ (no. of points generated inside the square)    pi = 4 * circle_points / square_points ##    print(rand_x, rand_y, circle_points, square_points, "-", pi)# print("\n") print("Final Estimation of Pi=", pi)

## C#

 // C# program for estimation of Pi using Monte// Carlo Simulationusing System;using System.Collections.Generic; class GFG {   // Defines precision for x and y values. More the  // interval, more the number of significant digits  static int INTERVAL = 10000;   // Driver code  public static void Main(string[] args)  {     // Instantiate random number generator using    // system-supplied value as seed    var rand = new Random();    double rand_x, rand_y, origin_dist, pi = 0;    int circle_points = 0, square_points = 0;     // Total Random numbers generated = possible x    // values * possible y values    for (int i = 0; i < (INTERVAL * INTERVAL); i++) {       // Randomly generated x and y values in the      // range [-1,1]      rand_x = (double)(rand.Next() % (INTERVAL + 1))        / INTERVAL;      rand_y = (double)(rand.Next() % (INTERVAL + 1))        / INTERVAL;       // Distance between (x, y) from the origin      origin_dist = rand_x * rand_x + rand_y * rand_y;       // Checking if (x, y) lies inside the define      // circle with R=1      if (origin_dist <= 1)        circle_points++;       // Total number of points generated      square_points++;       // estimated pi after this iteration      pi = ((4.0 * circle_points) / square_points);       // For visual understanding (Optional)      // System.out.println(rand_x+" "+rand_y+"      // "+circle_points+" "+square_points+" - "+pi);    }     // Final Estimated Value    Console.WriteLine("Final Estimation of Pi = " + pi);  }} // This code is contributed by phasing17

## Javascript

 /* JavaScript program for estimation of Pi using Monte   Carlo Simulation */ // Defines precision for x and y values. More the// interval, more the number of significant digitslet INTERVAL = 10000     let interval,    i;let rand_x, rand_y, origin_dist, pi;let circle_points = 0, square_points = 0; // Total Random numbers generated = possible x// values * possible y valuesfor (i = 0; i < (INTERVAL * INTERVAL); i++) {     // Randomly generated x and y values    rand_x = (Math.random() * (INTERVAL)) / INTERVAL;    rand_y = (Math.random() * (INTERVAL)) / INTERVAL;     // Distance between (x, y) from the origin    origin_dist = rand_x * rand_x + rand_y * rand_y;     // Checking if (x, y) lies inside the define    // circle with R=1    if (origin_dist <= 1)        circle_points++;     // Total number of points generated    square_points++;     // estimated pi after this iteration    pi = (4 * circle_points) / square_points;     // For visual understanding (Optional)    // console.log(rand_x,  rand_y , circle_points,    // square_points, "-", pi)} // Final Estimated Valueconsole.log("\nFinal Estimation of Pi = " + pi); // This code is contributed by phasing17

Output:

Final Estimation of Pi = 3.16116

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