Monte Carlo simulation basics in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of a simple Monte Carlo estimation of π

What is the output of this code that estimates π using Monte Carlo simulation with 1000 points?

NumPy

import numpy as np
np.random.seed(0)
points = 1000
x = np.random.uniform(-1, 1, points)
y = np.random.uniform(-1, 1, points)
inside_circle = (x**2 + y**2) <= 1
pi_estimate = 4 * np.sum(inside_circle) / points
print(round(pi_estimate, 2))

A 3.14

B 2.56

C 3.00

D 4.00

💡 Hint

Think about how many points fall inside the unit circle compared to total points.

The code generates 1000 random points in a square from -1 to 1 on both axes. It counts how many fall inside the unit circle. The ratio multiplied by 4 estimates π. With seed 0, the estimate rounds to 3.14.

❓ data_output

intermediate

Number of points inside circle in Monte Carlo simulation

After running this Monte Carlo simulation code with 5000 points, what is the number of points inside the unit circle?

NumPy

import numpy as np
np.random.seed(1)
points = 5000
x = np.random.uniform(-1, 1, points)
y = np.random.uniform(-1, 1, points)
inside_circle = (x**2 + y**2) <= 1
count_inside = np.sum(inside_circle)
print(count_inside)

A 5000

B 4000

C 3932

D 2500

💡 Hint

Count how many points satisfy the circle condition.

With seed 1 and 5000 points, the sum of points inside the circle is 3932.

❓ visualization

advanced

Visualizing Monte Carlo points inside and outside the circle

Which option shows the correct scatter plot of 1000 Monte Carlo points with points inside the unit circle in blue and outside in red?

NumPy

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(2)
points = 1000
x = np.random.uniform(-1, 1, points)
y = np.random.uniform(-1, 1, points)
inside_circle = (x**2 + y**2) <= 1
plt.figure(figsize=(5,5))
plt.scatter(x[inside_circle], y[inside_circle], color='blue', s=5, label='Inside Circle')
plt.scatter(x[~inside_circle], y[~inside_circle], color='red', s=5, label='Outside Circle')
plt.legend()
plt.title('Monte Carlo Points Inside and Outside Unit Circle')
plt.show()

A Scatter plot with points outside circle in blue only

B Red points inside circle, blue points outside, no legend

C All points in one color, no distinction

D Blue points inside circle, red points outside, circle boundary visible

💡 Hint

Check colors and labels for inside vs outside points.

The code colors points inside the circle blue and outside red, with a legend and title. Option D matches this exactly.

🧠 Conceptual

advanced

Understanding Monte Carlo simulation error behavior

Which statement correctly describes how the error in Monte Carlo estimation changes as the number of random points increases?

A Error decreases proportionally to 1 divided by the square root of the number of points

B Error decreases linearly with the number of points

C Error increases as more points are added

D Error remains constant regardless of the number of points

💡 Hint

Think about how randomness averages out with more samples.

Monte Carlo error typically decreases with the square root of the number of samples, meaning more points reduce error but with diminishing returns.

🔧 Debug

expert

Identify the error in Monte Carlo simulation code

What error will this code raise when run, and why?

NumPy

import numpy as np
np.random.seed(3)
points = 1000
x = np.random.uniform(-1, 1, points)
y = np.random.uniform(-1, 1, points)
inside_circle = (x**2 + y**2) <= 1
count_inside = np.sum(inside_circle)
print(count_inside)

A SyntaxError due to missing parentheses

B No error, prints the count of points inside circle

C TypeError because y is a float, not an array, so x**2 + y**2 fails

D ValueError because points is not an integer

💡 Hint

Check the shape and type of y compared to x.

y is generated as an array like x. There is no error, and the code prints the count of points inside the circle.