Hassan Ijaz
Ai, Web & Design
Probability axioms and basic rules
Interactive probability calculator with visual Venn diagrams that update in real-time as users adjust probabilities, showing how addition and multiplication rules work
Concept Overview
Probability theory provides a mathematical framework for quantifying uncertainty. At its foundation are three axioms established by Andrey Kolmogorov that define what we mean by probability.
Basic Notation & Concepts
- P(A): The probability that event A occurs
- P(A ∩ B) or P(A AND B): The probability that both A and B occur together
- P(A ∪ B) or P(A OR B): The probability that at least one of A or B occurs
- P(A|B): The probability of A occurring, given that B has already occurred (conditional probability)
- P(A') or P(not A): The probability that A does not occur (complement)
Examples:
- P(Rain) = 0.3 means 30% chance of rain
- P(Rain ∩ Cold) = 0.1 means 10% chance of both rain AND cold
- P(Rain ∪ Cold) = 0.5 means 50% chance of rain OR cold (or both)
- P(Rain|Cold) = 0.4 means 40% chance of rain IF it's already cold
Key Relationships Between Events
- Mutually Exclusive Events: Cannot happen togetherP(A ∩ B) = 0 (Example: Rolling a 1 AND a 6 on a single die - impossible!)
- Independent Events: One event doesn't affect the other's probabilityP(A|B) = P(A) - because B does not have any effect of A
P(A ∩ B) = P(A) × P(B) (Example: Coin flips - first flip doesn't affect second)Note: Independent events CAN overlap! If P(A) = 0.5, P(B) = 0.4, and they're independent, then P(A ∩ B) = 0.5 × 0.4 = 0.2 (they overlap 20% of the time) - Dependent Events: One event affects the other's probabilityP(A|B) ≠ P(A) (Example: Drawing cards without replacement - first card affects second)
The Three Axioms
- Non-negativity: For any event A, P(A) ≥ 0
- Normalization: The probability of the entire sample space S is P(S) = 1
- Additivity: For mutually exclusive events A and B (where P(A ∩ B) = 0), P(A ∪ B) = P(A) + P(B)
Key Rules Derived from Axioms
- General Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) (works for any events)
- Complement Rule: P(A') = 1 - P(A)
- Multiplication Rule: P(A ∩ B) = P(A) × P(B|A)
Note: When events are mutually exclusive (P(A ∩ B) = 0), the general addition rule simplifies to the axiom form: P(A ∪ B) = P(A) + P(B).
Use the interactive visualization below to explore how these rules work in practice. Adjust the probability sliders and observe how the Venn diagram updates in real-time.
Interactive Visualization
Loading interactive visualization...
Interactive probability calculator with visual Venn diagrams that update in real-time as users adjust probabilities, showing how addition and multiplication rules work