Learn Axioms and Properties of Probability

Section 1. Chapter 4

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To understand probability theory, you must first master its foundational rules. The three axioms of probability, established by Andrey Kolmogorov, define the logical structure for assigning probabilities to events. These axioms are:

Non-negativity: for any event $A$ , the probability of $A$ is always greater than or equal to zero: $P(A) ≥ 0;$
Normalization (certainty): the probability of the sample space $S$ (the event that something in the space occurs) is exactly one: $P(S) = 1;$
Additivity (disjoint events): for any two mutually exclusive (disjoint) events $A$ and $B$ (meaning they cannot both occur at the same time), the probability that either $A$ or $$Bv occurs is the sum of their individual probabilities: $P(A ∪ B) = P(A) + P(B)` if `A ∩ B = ∅.$

From these axioms, several important properties are derived. The complement rule states that the probability of an event not occurring is one minus the probability that it does occur:
$P(A^c) = 1 - P(A)$ , where $A^c$ is the complement of event $$Av.

Another key property is the addition rule for any two events, not necessarily disjoint:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B);

This formula accounts for the possibility that $A$ and $B$ may overlap, ensuring you do not double-count the intersection.

Understanding these axioms and properties is crucial for analyzing any random process or experiment.


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# Demonstrating the complement and addition rules with simple events

# Suppose you have a fair six-sided die.
# Let A = "rolling an even number"
# Let B = "rolling a number greater than 4"

# The sample space S = {1, 2, 3, 4, 5, 6}
# Event A = {2, 4, 6} -> P(A) = 3/6 = 0.5
# Event B = {5, 6} -> P(B) = 2/6 ≈ 0.333

P_A = 3 / 6
P_B = 2 / 6

# Intersection: A ∩ B = {6} -> P(A ∩ B) = 1/6 ≈ 0.167
P_A_and_B = 1 / 6

# Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
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)

# Complement rule: P(A^c) = 1 - P(A)
P_not_A = 1 - P_A
print("P(not A):", P_not_A)

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Section 1. Chapter 4

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