Learn Implementing Probability Basics in Python

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:

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

The intersection $A \cap B = \{2\}$ → common element.
The union $A \cup B = \{1,2,3,4,6\}$ → all elements in A or B.

Calculating Probabilities


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# 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)

We use the formulas:

$P(A) = \frac{\raisebox{1pt}{$|A|$}}{\raisebox{-1pt}{$6$}} = \frac{\raisebox{1pt}{$3$}}{\raisebox{-1pt}{$6$}}$ ;
$P(B) = \frac{\raisebox{1pt}{$|B|$}}{\raisebox{-1pt}{$6$}} = \frac{\raisebox{1pt}{$3$}}{\raisebox{-1pt}{$6$}}$ ;
$P(A \cap B) = \frac{\raisebox{1pt}{$|A \cap B|$}}{\raisebox{-1pt}{$6$}} = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$6$}}$ ;
$P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{\raisebox{1pt}{$5$}}{\raisebox{-1pt}{$6$}}$ .

Additional Set Details


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

print(only_A)
print(only_B)

Elements only in A: {1, 3};
Elements only in B: {4, 6}.

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Section 5. Chapter 2

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