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Mathematics for Data Science

bookImplementing Probability Basics in Python

Probability concepts are the foundation of analyzing uncertain events.
Here we learn how to compute union and intersection using a simple dice example.

Defining Sample Space and Events

# Small numbers on a die
A = {1, 2, 3}

# Even numbers on a die  
B = {2, 4, 6}  

die_outcomes = 6

Here we define:

  • A={1,2,3}A = \{1,2,3\} representing "small" outcomes;
  • B={2,4,6}B = \{2,4,6\} representing "even" outcomes.

The total number of die outcomes is 6.

Performing Set Operations


              
12345678

            
# Small numbers on a die
A = {1, 2, 3}
# Even numbers on a die  
B = {2, 4, 6}  
die_outcomes = 6

print(f'A and B = {A & B}')  # {2}
print(f'A or B = {A | B}')   # {1, 2, 3, 4, 6}
copy
  • The intersection A∩B={2}A \cap B = \{2\} → common element.
  • The union A∪B={1,2,3,4,6}A \cup B = \{1,2,3,4,6\} → all elements in A or B.

Calculating Probabilities


              
123456789101112131415161718

            
# Small numbers on a die
A = {1, 2, 3}
# Even numbers on a die  
B = {2, 4, 6}  
die_outcomes = 6

A_and_B = A & B  # {2}
A_or_B = A | B   # {1, 2, 3, 4, 6}

P_A = len(A) / die_outcomes
P_B = len(B) / die_outcomes
P_A_and_B = len(A_and_B) / die_outcomes
P_A_or_B = P_A + P_B - P_A_and_B

print("P(A) =", P_A)
print("P(B) =", P_B)
print("P(A ∩ B) =", P_A_and_B)
print("P(A ∪ B) =", P_A_or_B)
copy

We use the formulas:

  • P(A)=∣A∣6=36P(A) = \frac{|A|}{6} = \tfrac{3}{6};
  • P(B)=∣B∣6=36P(B) = \frac{|B|}{6} = \tfrac{3}{6};
  • P(A∩B)=∣A∩B∣6=16P(A \cap B) = \frac{|A \cap B|}{6} = \tfrac{1}{6};
  • P(A∪B)=P(A)+P(B)−P(A∩B)=56P(A \cup B) = P(A) + P(B) - P(A \cap B) = \tfrac{5}{6}.

Additional Set Details


              
12345

            
only_A = A - B   # {1, 3}
only_B = B - A   # {4, 6}

print(only_A)
print(only_B)
copy
  • Elements only in A: {1, 3};
  • Elements only in B: {4, 6}.

1. What is the output of this code?

2. What does this line compute?

3. What is the result of this code?

4. Which line correctly calculates the union probability using Python?

5. What does this code return?

question mark

What is the output of this code?

Select the correct answer

question mark

What does this line compute?

Select the correct answer

question mark

What is the result of this code?

Select the correct answer

question mark

Which line correctly calculates the union probability using Python?

Select the correct answer

question mark

What does this code return?

Select the correct answer

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Секція 5. Розділ 2

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bookImplementing Probability Basics in Python

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Probability concepts are the foundation of analyzing uncertain events.
Here we learn how to compute union and intersection using a simple dice example.

Defining Sample Space and Events

# Small numbers on a die
A = {1, 2, 3}

# Even numbers on a die  
B = {2, 4, 6}  

die_outcomes = 6

Here we define:

  • A={1,2,3}A = \{1,2,3\} representing "small" outcomes;
  • B={2,4,6}B = \{2,4,6\} representing "even" outcomes.

The total number of die outcomes is 6.

Performing Set Operations


              
12345678

            
# Small numbers on a die
A = {1, 2, 3}
# Even numbers on a die  
B = {2, 4, 6}  
die_outcomes = 6

print(f'A and B = {A & B}')  # {2}
print(f'A or B = {A | B}')   # {1, 2, 3, 4, 6}
copy
  • The intersection A∩B={2}A \cap B = \{2\} → common element.
  • The union A∪B={1,2,3,4,6}A \cup B = \{1,2,3,4,6\} → all elements in A or B.

Calculating Probabilities


              
123456789101112131415161718

            
# Small numbers on a die
A = {1, 2, 3}
# Even numbers on a die  
B = {2, 4, 6}  
die_outcomes = 6

A_and_B = A & B  # {2}
A_or_B = A | B   # {1, 2, 3, 4, 6}

P_A = len(A) / die_outcomes
P_B = len(B) / die_outcomes
P_A_and_B = len(A_and_B) / die_outcomes
P_A_or_B = P_A + P_B - P_A_and_B

print("P(A) =", P_A)
print("P(B) =", P_B)
print("P(A ∩ B) =", P_A_and_B)
print("P(A ∪ B) =", P_A_or_B)
copy

We use the formulas:

  • P(A)=∣A∣6=36P(A) = \frac{|A|}{6} = \tfrac{3}{6};
  • P(B)=∣B∣6=36P(B) = \frac{|B|}{6} = \tfrac{3}{6};
  • P(A∩B)=∣A∩B∣6=16P(A \cap B) = \frac{|A \cap B|}{6} = \tfrac{1}{6};
  • P(A∪B)=P(A)+P(B)−P(A∩B)=56P(A \cup B) = P(A) + P(B) - P(A \cap B) = \tfrac{5}{6}.

Additional Set Details


              
12345

            
only_A = A - B   # {1, 3}
only_B = B - A   # {4, 6}

print(only_A)
print(only_B)
copy
  • Elements only in A: {1, 3};
  • Elements only in B: {4, 6}.

1. What is the output of this code?

2. What does this line compute?

3. What is the result of this code?

4. Which line correctly calculates the union probability using Python?

5. What does this code return?

question mark

What is the output of this code?

Select the correct answer

question mark

What does this line compute?

Select the correct answer

question mark

What is the result of this code?

Select the correct answer

question mark

Which line correctly calculates the union probability using Python?

Select the correct answer

question mark

What does this code return?

Select the correct answer

Все було зрозуміло?

Як ми можемо покращити це?

Дякуємо за ваш відгук!

Секція 5. Розділ 2
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