Monte Carlo simulation basics in NumPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to generate 1000 random numbers between 0 and 1 using NumPy.

NumPy

import numpy as np
random_numbers = np.random.[1](1000)

Drag options to blanks, or click blank then click option'

Arandint

Brand

Crandom_sample

Drandom

Attempts:

3 left

💡 Hint

Common Mistakes

Using randint which generates integers, not floats.

Using random_sample which is an alias but not the most common.

The np.random.rand function generates random numbers between 0 and 1.

2fill in blank

medium

Complete the code to estimate the probability of getting a value less than 0.5 from the random numbers.

NumPy

prob = np.mean(random_numbers [1] 0.5)

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A==

B>

C<

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using greater than operator which counts wrong values.

Using equality which is too strict.

We check which numbers are less than 0.5 and take the mean to estimate the probability.

3fill in blank

hard

Fix the error in the code to simulate rolling a six-sided die 1000 times.

NumPy

rolls = np.random.randint(1, [1], size=1000)

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A5

B6

C0

D7

Attempts:

3 left

💡 Hint

Common Mistakes

Using 6 as upper bound excludes 6 from results.

Using 0 as upper bound causes an error.

The upper bound in randint is exclusive, so to include 6, we use 7.

4fill in blank

hard

Fill both blanks to create a dictionary with counts of rolls greater than 4 and less than or equal to 4.

NumPy

counts = { 'gt_4': np.sum(rolls [1] 4), 'le_4': np.sum(rolls [2] 4) }

Drag options to blanks, or click blank then click option'

A>

B<=

C<

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using < instead of <= for 'le_4'.

Mixing up operators causing wrong counts.

We count rolls greater than 4 and rolls less than or equal to 4 using > and <= operators.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that counts how many rolls are equal to each face from 1 to 6.

NumPy

face_counts = { [1]: np.sum(rolls [2] [3]) for [1] in range(1, 7) }

Drag options to blanks, or click blank then click option'

Aface

B==

Drolls

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'rolls' as key instead of 'face'.

Using assignment '=' instead of comparison '=='.

The dictionary comprehension uses face as key, checks where rolls equal face with ==, and sums those occurrences.

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