Question

This problem is also a Monte Carlo simulation, but this time in the continuous domain: must use the following fact: a circle inscribed in a unit square

has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.

Therefore, if you generate num_trials random points in the unit square, and count how many land inside the circle, you can calculate an approximation of ?

For this problem, you must create code in python

(B) Without drawing the diagram, calculate the value of ? you would get from 10⁵ trials.

(C) After completing (B), try to get a more accurate value for ? by increasing the number of trials.The results will depend on your machine

Answer 1

import numpy as np
import matplotlib.pyplot as pl

trials = 105
counts = 0
for i in range(trials):
          x = np.random.random()
          y = np.random.random()
          x2 = x**2
          y2 = y**2
          x_y = x2 + y2
          dxy = np.sqrt(x_y)
          if dxy <= 1:
                    counts = counts + 1

print ("number of trial =" ,trials ,"pi was approximated as ::",4*counts/trials)

you can chane the value of trials to larger number to see improvement in pi

Note that function name is approx_pi.py