Let S be the square centered at the origin with sides of length 2, and C...

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Basic Math

Let S be the square centered at the origin with sides of length2, and C be the unit circle centered at the origin.
(a) If you randomly throw a point on S, what is the probabilitythat it will lie in C?
Ans: 0.785
(b) Describe how you could use simulation to estimate theprobability in part (a).
(c) How can you use simulation to estimate a?
For part b and c, there maybe a need to generate randomvariables for the simulation by Box-Muller, Accept/Reject orimportance sampling. Any help on b and c will be appreciated !

Answer & Explanation Solved by verified expert

3.6 Ratings (643 Votes)

probability = area of circle/ area of square = pi/4 = 0.785

we can generate two random random numbers x and y from 0 to 1

and if sqrt(x^2+ y^2) < 1 , then we call it success , we can simulate this or many trials

code in python

import numpy as np
import matplotlib.pyplot as pl

trials = 10000
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 ,",required probability is ::",counts/trials)

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