Question

1. Consider a dartboard consisting of a circle of radius r inscribed in a square of side length 2r. Assume that there is some algorithm DART that randomly and uniformly throws a dart at this board, returning 1 if the dart lands within the circle and returning 0 otherwise.

Answer 1

//The solution of the above problem is attached below:

//CPP CODE-

#include <iostream>

#include <cmath>

#include <cstdlib>

#include <ctime>

#include <iomanip>

#include <string>

using namespace std;

int main()

{

const double PI = 3.141592653589793;

static double x_cor = 0;

static double y_cor = 0;

static double x_sum = 0;

static double y_sum = 0;

double x_y_total = 0;

long int num_darts = 0;

long int num_hits = 0;

long int num_misses = 0;

long double pi_approx = 0;

srand(time(0));

while ( num_darts != 0 ) // Start of while loop

{

cout << " How many times do you want to throw the darts: ";

cin >> num_darts;

cout << endl;

if ( num_darts != 0 )

{

for ( int i = 0; i < num_darts; i++) // Start of for loopa inside the while loop

{

x_cor = static_cast<double>(rand())/RAND_MAX;

y_cor = static_cast<double>(rand())/RAND_MAX;

x_sum = (x_cor * x_cor);

y_sum = (y_cor * y_cor);

x_y_total = x_sum + y_sum;

if (x_y_total < 1)

{

num_hits++;

}

else

{

num_misses++;

}

}

} // end of if loop

pi_approx = (num_hits / num_darts);

} // end of while loop

cout << " The value is " << pi_approx << endl;

return 0;

}