Question

Python

Problem 4. Estimate pi version 2: write a program that will quickly estimate pi to a precision of 1e-4 using a monte carlo approach. Your program should employ a loop where each iteration produces a new x,y pair. One will then determine whether the x,y pair lies inside or outside the circle. (see below) Since all the points will land within the square shown above but only a fraction will land within the circle, we can estimate pi by relating the area of the quarter circle shown above to the area of the square. As we randomly choose points, we can determine if they fall within the circle or not and form a ratio as shown below: AreaQtrcircle / AreaSqaure = πr 2 / 4r 2 = π / 4 = Points in Circle / All Points, π ≅ 4* Points in Circle / All Points.

Your loop should continue until the absolute value of the difference between the current estimate and the previous estimate is less than 1e-4. The program should output the estimate of pi and the number of iterations (points) needed to reach the desired level of precision. Run your program several times to see if the number of points changes. Note any observations in the comments.

Answer 1

import random

prevEstimate = None
currentEstimate = None

iterations = 0
pointsInCircle = 0
pointsInSquare = 0

while True:
        (x, y) = (random.random(), random.random())
        iterations += 1

        if x*x + y*y <= 1:
                pointsInCircle += 1
        pointsInSquare += 1     

        # min 100 iterations should be done.
        currentEstimate = 4.0 * pointsInCircle / pointsInSquare
        if prevEstimate is None or iterations <= 100:
                prevEstimate = currentEstimate
        elif abs(prevEstimate - currentEstimate) < 10**-4:
                break
        else:
                prevEstimate = currentEstimate

print('Value:', currentEstimate)
print('Iterations', iterations)

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