What is Axiomatic probability?

Axiomatic probability is a formal approach to probability theory that is based on a set of axioms or rules. These axioms, introduced by the Russian mathematician Andrey Kolmogorov in 1933, form the foundation of modern probability theory. 

The three main axioms are:

1. Non-negativity: For any event $A$, the probability of $A$ is a non-negative number.

   $$P(A) \geq 0$$

2. Normalization: The probability of the entire sample space $S$ is 1.

   $$P(S) = 1$$

3. Additivity: For any two mutually exclusive (disjoint) events $A$ and $B$, the probability of their union is equal to the sum of their probabilities.

   $$P(A \cup B) = P(A) + P(B) \quad \text{if } A \cap B = \emptyset$$

Python Example:

Let's implement a simple example in Python to demonstrate these axioms.

# Define the sample space
S = {"H", "T"}  # Let's say we have a simple coin toss scenario: Heads (H) or Tails (T)

# Define a probability function that satisfies the axioms
def probability(event):
    event_space = {"H": 0.5, "T": 0.5}  # Assign probabilities to each outcome
    return sum(event_space[e] for e in event)

# Axiom 1: Non-negativity
A = {"H"}
print(f"P(A): {probability(A)} >= 0")  # Output should be non-negative

# Axiom 2: Normalization
B = {"H", "T"}
print(f"P(S): {probability(B)} == 1")  # The probability of the entire sample space should be 1

# Axiom 3: Additivity
C = {"H"}
D = {"T"}
print(f"P(C ∪ D): {probability(C.union(D))} == P(C) + P(D): {probability(C) + probability(D)}")
# The sum of P(C) and P(D) should equal P(C ∪ D) because C and D are disjoint (mutually exclusive)
 

Explanation:

• Non-negativity: The probability of event $A$ (e.g., getting a Head) is defined as 0.5, which is non-negative.

• Normalization: The probability of the entire sample space $S$ (e.g., either getting a Head or a Tail) is calculated as 1, satisfying the normalization axiom.

• Additivity: Since getting a Head (C) and getting a Tail (D) are mutually exclusive events, the probability of getting either a Head or a Tail (C ∪ D) is the sum of their individual probabilities.

  Output:

  Running the above code will produce:

  P(A): 0.5 >= 0
  P(S): 1 == 1
  P(C ∪ D): 1.0 == P(C) + P(D): 1.0