Welcome to the foundational concepts of probability and statistics, crucial for success in competitive exams like the Society of Actuaries (SOA) actuarial exams. This module focuses on the bedrock of probability theory: the Axioms of Probability. Understanding these axioms is essential for building a robust framework for analyzing uncertain events.

Probability is a measure of the likelihood that an event will occur. It's a numerical value between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. In actuarial science, probability is fundamental to risk assessment and financial modeling.

Developed by Andrey Kolmogorov, these axioms form the mathematical foundation of probability theory. They are simple yet powerful, ensuring consistency and logical coherence in our probabilistic reasoning.

From these three simple axioms, we can derive many important properties of probability, such as:

• The probability of an impossible event is 0 (P(∅) = 0).
• For any event E, 0 ≤ P(E) ≤ 1.
• For any two events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B) (the addition rule).

In actuarial exams, you'll encounter problems that require you to apply these axioms directly or indirectly. For instance, when dealing with insurance claims, understanding the probability of different claim scenarios (mutually exclusive events) and the overall probability of a policy being active (sample space) is crucial. Mastering these foundational axioms will provide a solid base for more complex topics like conditional probability, random variables, and distributions.