1.1 Probability axioms and properties

Probability axioms and properties form the foundation of actuarial mathematics. They provide a framework for quantifying uncertainty and assessing risk in various scenarios. Understanding these concepts is crucial for actuaries to make informed decisions and develop accurate models.

These fundamental principles enable actuaries to calculate probabilities, analyze complex events, and update their assessments based on new information. Mastering these concepts is essential for tackling more advanced topics in actuarial science and applying them to real-world problems.

Probability basics

• Probability is a fundamental concept in actuarial mathematics used to quantify the likelihood of events occurring
• Understanding probability is essential for actuaries to assess risk, make informed decisions, and price insurance products

Sample space and events

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• Sample space ($\Omega$) represents the set of all possible outcomes of an experiment or random process
• An event is a subset of the sample space, denoting a collection of outcomes of interest (rolling an even number on a die)
• Events can be simple or compound, depending on whether they consist of a single outcome or multiple outcomes
• The empty set ($\emptyset$) and the sample space itself are also considered events

Probability of an event

• Probability ($P(A)$) measures the likelihood of an event $A$ occurring, expressed as a value between 0 and 1
• For a finite sample space, the probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes
• Probability can be determined using classical, empirical, or subjective approaches depending on the context and available information

Axioms of probability

• The axioms of probability provide a rigorous foundation for the mathematical theory of probability
• These axioms ensure that probability measures are consistent, coherent, and adhere to certain desirable properties

Non-negativity

• The probability of any event $A$ is always non-negative: $P(A) \geq 0$
• This axiom guarantees that probabilities cannot be negative, as it would be counterintuitive and inconsistent with the notion of likelihood

Probability of sample space

• The probability of the entire sample space $\Omega$ is equal to 1: $P(\Omega) = 1$
• This axiom reflects the idea that the sample space encompasses all possible outcomes, and one of them must occur with certainty

Countable additivity

• For any countable sequence of mutually exclusive events $A_1, A_2, \ldots$, the probability of their union is equal to the sum of their individual probabilities: $P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$
• This axiom allows for the calculation of probabilities of compound events by breaking them down into simpler, mutually exclusive components

Consequences of axioms

• The axioms of probability lead to several important consequences and properties that facilitate the calculation and manipulation of probabilities

Probability of empty set

• The probability of the empty set $\emptyset$ is always 0: $P(\emptyset) = 0$
• This follows from the non-negativity and countable additivity axioms, as the empty set contains no outcomes

Probability of complement

• The probability of the complement of an event $A$, denoted as $A^c$ or $\overline{A}$, is given by: $P(A^c) = 1 - P(A)$
• This property allows for the calculation of the probability of an event not occurring, given the probability of the event occurring

Monotonicity of probability

• For any two events $A$ and $B$, if $A \subseteq B$, then $P(A) \leq P(B)$
• This property states that if an event $A$ is a subset of another event $B$, then the probability of $A$ cannot exceed the probability of $B$

Properties of probability

• Several important properties of probability arise from the axioms and their consequences, enabling the manipulation and calculation of probabilities in various scenarios

Inclusion-exclusion for two events

• For any two events $A$ and $B$, the probability of their union is given by: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• This property accounts for the double-counting of outcomes that belong to both events $A$ and $B$

Inclusion-exclusion for multiple events

• The inclusion-exclusion principle generalizes to $n$ events $A_1, A_2, \ldots, A_n$: $P(\bigcup_{i=1}^{n} A_i) = \sum_{i=1}^{n} P(A_i) - \sum_{i<j} P(A_i \cap A_j) + \sum_{i<j<k} P(A_i \cap A_j \cap A_k) - \ldots + (-1)^{n+1} P(A_1 \cap A_2 \cap \ldots \cap A_n)$
• This formula alternates between adding and subtracting the probabilities of intersections of events to avoid multiple counting

Bonferroni inequalities

• The Bonferroni inequalities provide upper and lower bounds for the probability of the union of $n$ events:
  • Lower bound: $P(\bigcup_{i=1}^{n} A_i) \geq \sum_{i=1}^{n} P(A_i) - \sum_{i<j} P(A_i \cap A_j)$
  • Upper bound: $P(\bigcup_{i=1}^{n} A_i) \leq \sum_{i=1}^{n} P(A_i)$
• These inequalities are useful when the exact probabilities of intersections are unknown or difficult to calculate

Continuity of probability

• If a sequence of events $A_1, A_2, \ldots$ converges to an event $A$ (i.e., $\lim_{n \to \infty} A_n = A$), then the probability of $A_n$ converges to the probability of $A$: $\lim_{n \to \infty} P(A_n) = P(A)$
• This property ensures that probability measures are continuous, allowing for the approximation of probabilities using limiting processes

Conditional probability

• Conditional probability measures the probability of an event $A$ occurring given that another event $B$ has already occurred, denoted as $P(A|B)$
• Conditional probability is a fundamental concept in actuarial mathematics, as it allows for the updating of probabilities based on new information or observations

Definition and formula

• The conditional probability of event $A$ given event $B$ is defined as: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) > 0$
• This formula calculates the probability of the intersection of events $A$ and $B$, relative to the probability of event $B$

Law of total probability

• The law of total probability states that for a partition of the sample space $\{B_1, B_2, \ldots, B_n\}$, the probability of an event $A$ can be calculated as: $P(A) = \sum_{i=1}^{n} P(A|B_i) \cdot P(B_i)$
• This law allows for the calculation of the probability of an event by considering all possible mutually exclusive and exhaustive scenarios

Bayes' theorem

• Bayes' theorem describes the relationship between conditional probabilities and provides a way to update probabilities based on new evidence: $P(B_i|A) = \frac{P(A|B_i) \cdot P(B_i)}{\sum_{j=1}^{n} P(A|B_j) \cdot P(B_j)}$
• This theorem is essential in actuarial applications, such as updating risk assessments based on new data or observations

Independence of events

• Two events $A$ and $B$ are considered independent if the occurrence of one event does not affect the probability of the other event occurring
• Independence is a crucial concept in probability theory and actuarial mathematics, as it simplifies calculations and allows for the modeling of complex systems

Definition of independence

• Events $A$ and $B$ are independent if and only if: $P(A \cap B) = P(A) \cdot P(B)$
• This definition states that the probability of the intersection of independent events is equal to the product of their individual probabilities

Multiplication rule for independent events

• For a sequence of independent events $A_1, A_2, \ldots, A_n$, the probability of their intersection is given by: $P(A_1 \cap A_2 \cap \ldots \cap A_n) = P(A_1) \cdot P(A_2) \cdot \ldots \cdot P(A_n)$
• This rule allows for the calculation of the probability of multiple independent events occurring simultaneously

Pairwise vs mutual independence

• Pairwise independence refers to the situation where any two events in a collection are independent, but the entire collection may not be independent
• Mutual independence is a stronger condition, requiring that any subset of events in a collection is independent
• For a collection of events $\{A_1, A_2, \ldots, A_n\}$ to be mutually independent, every possible intersection of these events must satisfy the multiplication rule for independence