5.1 Fundamentals of Probability

Fundamentals of Probability

Quantitative Analysis opens with probability because every risk measure—VaR, expected shortfall, default likelihood, option pricing—is a statement about uncertain outcomes. GARP’s QA–1 reading expects fluency with events, conditioning, independence concepts, and Bayes’ rule before distribution catalogs begin.

Sample Space, Events, and Event Space

The sample space Ω is the set of all mutually exclusive, exhaustive outcomes of an experiment. Tossing a fair six-sided die once gives Ω = {1, 2, 3, 4, 5, 6}. For tomorrow’s equity return, Ω might be the continuum of real numbers (or a discretized grid in a model).

An event is any subset of Ω that we care about measuring. Examples: A = {even faces} = {2, 4, 6}; B = {loss greater than 2%}; C = {counterparty defaults within one year}. The collection of events we assign probabilities to is the event space (formally a σ-algebra). For FRM purposes, treat it as “the family of events the model recognizes,” closed under complement and countable unions so that “not A,” “A or B,” and “A and B” are always valid events when A and B are.

A probability measure P maps events to numbers in [0, 1] with three core properties:

1. Non-negativity: P(E) ≥ 0 for every event E.
2. Normalization: P(Ω) = 1.
3. Additivity for disjoint events: if A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B). For countably many pairwise-disjoint events, the same rule extends to infinite sums.

Immediate corollaries: P(∅) = 0; P(Aᶜ) = 1 − P(A); and for any A, B, the inclusion-exclusion identity P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Independent Versus Mutually Exclusive

Two events are mutually exclusive (disjoint) if they cannot happen together: A ∩ B = ∅, so P(A ∩ B) = 0. “Downgrade to CCC” and “upgrade to AA” in the same one-period rating migration for one issuer are mutually exclusive in a standard matrix.

Two events are independent if knowledge of one does not change the probability of the other: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A) when P(B) > 0. Independence is about information, not about impossibility of joint occurrence.

Critical trap: mutually exclusive events with positive individual probabilities are dependent, not independent. If P(A) > 0 and P(B) > 0 and A ∩ B = ∅, then P(A ∩ B) = 0 ≠ P(A)P(B). Exam writers love this confusion.

Independence extends to collections: A, B, C are mutually independent if every subset factors—P(A ∩ B) = P(A)P(B), P(A ∩ C) = P(A)P(C), P(B ∩ C) = P(B)P(C), and P(A ∩ B ∩ C) = P(A)P(B)P(C). Pairwise independence alone is not enough for the triple product.

Conditional Independence

Events A and B are conditionally independent given C if P(A ∩ B | C) = P(A|C)P(B|C). Unconditionally they may still be dependent because C induces correlation.

Risk example: two corporate bonds may look correlated because both load on the same credit cycle. Conditional on a latent factor F (economy state), default indicators D₁ and D₂ might be modeled as independent. That is the logic behind many one-factor credit-portfolio models: conditional independence given the systematic factor, unconditional dependence through the factor. FRM candidates should state which conditioning set is assumed whenever they claim “independent defaults.”

Discrete Probabilities and Elementary Calculations

When Ω is finite and outcomes are equally likely, P(A) = |A| / |Ω|. More generally, assign probabilities pᵢ to atoms ωᵢ with Σ pᵢ = 1, then P(A) = Σ_{ωᵢ ∈ A} pᵢ.

Worked discrete example

A trading desk faces three scenarios for next quarter with probabilities 0.50 (base), 0.30 (mild stress), and 0.20 (severe stress). Event L = “liquidity buffer breached” occurs in mild stress with probability 0.40 conditional on that scenario and in severe stress with probability 0.75; it never occurs in the base case. Then:

P(L) = 0.50·0 + 0.30·0.40 + 0.20·0.75 = 0 + 0.12 + 0.15 = 0.27.

This is the law of total probability: partition Ω into scenarios Sⱼ and write P(L) = Σ P(L|Sⱼ)P(Sⱼ).

Conditional Versus Unconditional Probability

Unconditional (marginal) probability P(A) is the probability before using a specific piece of information. Conditional probability updates given that B occurred:

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

Rearrangement: P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A).

Risk managers live in conditional space: P(default | downgrade), P(loss > x | VIX spike), P(breach | model break). Reporting only unconditional averages can hide that a book is safe in calm markets and fragile in stress.

From the liquidity example, P(severe | L) is not 0.20. Using Bayes below, most breaches come from severe and mild stress in proportion to their joint mass.

Bayes’ Rule

Bayes’ rule reverses conditioning:

P(Sⱼ|L) = P(L|Sⱼ)P(Sⱼ) / P(L).

Priors P(Sⱼ) combine with likelihoods P(L|Sⱼ) to produce posteriors P(Sⱼ|L).

Worked Bayes example (full numbers)

Using the same scenarios and P(L) = 0.27:

• P(base | L) = (0 · 0.50) / 0.27 = 0
• P(mild | L) = (0.40 · 0.30) / 0.27 = 0.12 / 0.27 ≈ 0.444
• P(severe | L) = (0.75 · 0.20) / 0.27 = 0.15 / 0.27 ≈ 0.556

Given a buffer breach, there is about a 55.6% chance the world is in severe stress and 44.4% it is mild stress—even though the unconditional prior on severe stress was only 20%. That is the exam point: data updates priors; rare scenarios with high likelihoods can dominate the posterior.

Credit-risk analogue: a sparse default sample still updates PD beliefs when combined with a prior from the rating system. Fraud detection, model validation “false positive” problems, and regime inference all use the same structure.

Putting the Rules to Work

When a FRM stem mentions two risks, ask: Are they exclusive, independent, or neither? When information arrives, switch from P(A) to P(A|B). When the question gives P(data|hypothesis) but asks for P(hypothesis|data), apply Bayes with a carefully computed denominator via the law of total probability. Master these mechanics and later chapters on distributions and multivariate dependence become applications rather than new languages.