Question

Calculate the value of π

During the lectures and labs, you have got familiar with calculating the value of using Monte Carlo method. Please write codes to define a function of the Monte Carlo simulation for this process using 10000 points (You could refer to the example codes on Canvas under the folder Lec10). Then please repeat this procedure for 4000 times, calculating the standard deviation, mean value and plot a histogram for the results. With the standard deviation, what's the probability of obtaining within the interval from л+σ to π-σ (σ is the standard deviation calculated out), if this procedure is repeated for another 4000 times? Is the result as expected?

Answer 1

To calculate the value of π using the Monte Carlo method, generate random points within a square and determine the proportion of points that fall within a quarter of a circle inscribed within that square. The ratio of the points inside the circle to the total number of points is proportional to the area of the circle relative to the square.

Step-by-step explanation:

To calculate the value of π using the Monte Carlo method, we can generate random points within a square and determine the proportion of points that fall within a quarter of a circle inscribed within that square. The ratio of the points inside the circle to the total number of points is proportional to the area of the circle relative to the square.

Here is a code to define a function for the Monte Carlo simulation:

import random

def monte_carlo_pi(num_points):
points_inside_circle = 0
for _ in range(num_points):
x = random.uniform(0, 1)
y = random.uniform(0, 1)
distance = x**2 + y**2
if distance <= 1:
points_inside_circle += 1
return 4 * points_inside_circle / num_points

To calculate the standard deviation and mean value, you can run the simulation for 4000 times and store the results. Then, calculate the standard deviation and mean value using the numpy library.