Monte Carlo simulation basics in NumPy

Introduction

Monte Carlo simulation helps us understand uncertain situations by trying many random examples. It shows possible outcomes and their chances.

Estimating the chance of winning a game with random moves.

Predicting future sales when customer behavior is uncertain.

Calculating the risk of investment portfolios with many unknown factors.

Finding the area of a shape that is hard to measure exactly.

Syntax

NumPy

import numpy as np

# Step 1: Generate many random samples
samples = np.random.random(size=N)

# Step 2: Apply a function or condition to samples
results = some_function(samples)

# Step 3: Calculate statistics like mean or probability
estimate = np.mean(results)

np.random.random(size=N) creates N random numbers between 0 and 1.

You can replace some_function with any calculation or test you want to simulate.

Examples

This example estimates the average of random numbers between 0 and 1, which should be close to 0.5.

NumPy

import numpy as np

N = 10000
samples = np.random.random(size=N)
estimate = np.mean(samples)
print(estimate)

This example estimates the value of π by checking how many random points fall inside a quarter circle.

NumPy

import numpy as np

N = 100000
samples_x = np.random.random(size=N)
samples_y = np.random.random(size=N)
inside_circle = (samples_x**2 + samples_y**2) < 1
pi_estimate = 4 * np.mean(inside_circle)
print(pi_estimate)

Sample Program

This program uses Monte Carlo simulation to estimate π. It randomly picks points in a square and counts how many fall inside a quarter circle. The ratio helps estimate π.

NumPy

import numpy as np

# Number of random trials
N = 100000

# Generate random points in the square [0,1] x [0,1]
x = np.random.random(size=N)
y = np.random.random(size=N)

# Check if points fall inside the quarter circle of radius 1
inside_circle = (x**2 + y**2) <= 1

# Estimate pi using the ratio of points inside the circle
pi_estimate = 4 * np.mean(inside_circle)

print(f"Estimated value of pi: {pi_estimate:.5f}")

Important Notes

More trials (larger N) give better estimates but take more time.

Monte Carlo simulation works well when exact math is hard or impossible.

Randomness means results will be slightly different each time you run the code.

Summary

Monte Carlo simulation uses random sampling to estimate results.

It is useful for problems with uncertainty or complex math.

Using numpy makes it easy to generate many random samples quickly.

Practice

1. What is the main purpose of a Monte Carlo simulation in data science?

easy

A. To create visual charts

B. To sort data quickly

C. To estimate results using random sampling

D. To clean missing data

Solution

1. Step 1: Understand Monte Carlo simulation basics

   Monte Carlo simulation uses random sampling to estimate results when exact calculation is hard.

2. Step 2: Identify the main purpose

   It helps estimate outcomes by simulating many random scenarios.

3. Final Answer:

   To estimate results using random sampling -> Option C

4. Quick Check:

   Monte Carlo = random sampling estimation [OK]

Hint: Monte Carlo = random sampling for estimation [OK]

Common Mistakes:

• Confusing Monte Carlo with data sorting
• Thinking it cleans data
• Assuming it creates charts

2. Which numpy function is commonly used to generate random samples for Monte Carlo simulations?

easy

A. numpy.random.rand()

B. numpy.array()

C. numpy.mean()

D. numpy.sort()

Solution

1. Step 1: Identify numpy functions for randomness

   numpy.random.rand() generates random numbers between 0 and 1, useful for sampling.

2. Step 2: Compare with other options

   numpy.array() creates arrays, numpy.mean() calculates average, numpy.sort() sorts data; none generate random samples.

3. Final Answer:

   numpy.random.rand() -> Option A

4. Quick Check:

   Random samples = numpy.random.rand() [OK]

Hint: Random samples come from numpy.random.rand() [OK]

Common Mistakes:

• Choosing numpy.array() instead of random function
• Confusing mean or sort with random sampling
• Using functions that don't generate randomness

3. What is the output of this code snippet?

import numpy as np
np.random.seed(0)
samples = np.random.rand(5)
mean_val = np.mean(samples)
print(round(mean_val, 2))

medium

A. 0.50

B. 0.57

C. 0.60

D. 0.45

Solution

1. Step 1: Generate 5 random samples with seed 0

   Using np.random.seed(0) fixes randomness. np.random.rand(5) produces [0.5488135, 0.71518937, 0.60276338, 0.54488318, 0.4236548].

2. Step 2: Calculate mean and round

   Sum = 2.83530423, mean = 0.567060846, round(mean_val, 2) = 0.57.

3. Final Answer:

   0.57 -> Option B

4. Quick Check:

   Mean of samples = 0.57 [OK]

Hint: Seed fixes random values; mean rounds to 0.57 [OK]

Common Mistakes:

• Not setting seed causing different outputs
• Rounding incorrectly
• Calculating sum or mean wrongly

4. Identify the error in this Monte Carlo simulation code:

import numpy as np
samples = np.random.rand(1000)
result = sum(samples) / 100
print(result)

medium

A. Using np.random.rand() instead of np.random.randint()

B. Not setting random seed

C. Missing np.mean() function

D. Dividing by 100 instead of number of samples

Solution

1. Step 1: Check calculation of average

   sum(samples) sums 1000 values, but dividing by 100 underestimates average.

2. Step 2: Correct divisor

   To get average, divide sum by 1000 (number of samples), not 100.

3. Final Answer:

   Dividing by 100 instead of number of samples -> Option D

4. Quick Check:

   Average = sum / number of samples [OK]

Hint: Divide sum by sample count, not fixed number [OK]

Common Mistakes:

• Dividing by wrong number
• Confusing random functions
• Ignoring seed importance

5. You want to estimate the probability that a random point inside a 1x1 square falls within a circle of radius 0.5 centered at (0.5, 0.5) using Monte Carlo simulation. Which numpy code correctly estimates this probability?

hard

A. points = np.random.rand(10000, 2)
dist = np.sqrt((points[:,0]-0.5)**2 + (points[:,1]-0.5)**2)
prob = np.sum(dist <= 0.5) / 10000
print(prob)

B. points = np.random.rand(10000)
dist = np.sqrt((points-0.5)**2)
prob = np.sum(dist <= 0.5) / 10000
print(prob)

C. points = np.random.rand(2, 10000)
dist = np.sqrt((points[:,0]-0.5)**2 + (points[:,1]-0.5)**2)
prob = np.sum(dist <= 0.5) / 10000
print(prob)

D. points = np.random.randint(0, 1, (10000, 2))
dist = np.sqrt((points[:,0]-0.5)**2 + (points[:,1]-0.5)**2)
prob = np.sum(dist <= 0.5) / 10000
print(prob)

Solution

1. Step 1: Generate 2D points inside 1x1 square

   np.random.rand(10000, 2) creates 10000 points with x,y in [0,1].

2. Step 2: Calculate distance from center (0.5,0.5)

   Distance formula sqrt((x-0.5)^2 + (y-0.5)^2) applied to each point.

3. Step 3: Count points inside circle and estimate probability

   Count points with distance ≤ 0.5, divide by total points for probability.

4. Final Answer:

   np.random.rand(10000, 2) with column indexing for coordinates -> Option A

5. Quick Check:

   2D points + distance check = probability [OK]

Hint: Use 2D random points and distance formula [OK]

Common Mistakes:

• Using 1D points instead of 2D
• Wrong shape for points array
• Using randint instead of rand for floats