Probability Fundamentals and Random Variables

You estimate a 2% one-day default rate on a counterparty. The portfolio holds 200 such names. The exam will ask you the probability of zero defaults, the expected number of defaults, and how that estimate flips when one default conditions on another. The math underneath every one of those questions is the probability machinery in this lesson.

The event space is the set of every distinct outcome a random experiment can produce. For a single bond over one year, the event space might be {default, no default}. For a credit migration model, it might be the seven rating buckets from AAA to default. Events are subsets of the event space: "downgrade" is the union of every outcome where the rating drops at least one notch.

Probabilities are non-negative numbers that sum to 1 across the event space. When you write $P(A)$, you are pulling the probability mass of every outcome inside subset $A$ and adding it.

KEY: Mutually exclusive events do not overlap, so their probabilities add: $P(A\cup B)=P(A)+P(B)$. When events can overlap, you subtract the intersection: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

Common mistakes

• Adding probabilities of overlapping events. $P(A\cup B)=P(A)+P(B)$ only when $A$ and $B$ are mutually exclusive. For overlapping events, subtract the intersection. Trap: a question asks the probability of "default OR downgrade" with $P(\text{default})=0.02$ and $P(\text{downgrade})=0.10$; the trap answer is 0.12, but if defaults are a subset of downgrades the right answer is 0.10.
• Confusing mutual exclusivity with independence. Mutually exclusive events with positive probabilities are dependent: knowing one happened tells you the other did not. Trap: "events are mutually exclusive, so they're independent" appears as a confident-sounding distractor.
• Forgetting the marginal in Bayes' rule. $P(A\mid B)=P(B\mid A)P(A)/P(B)$. The denominator must use the law of total probability: $P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)$. Trap: candidates compute the numerator and divide by $P(B\mid A)$ or by $P(A^c)$ instead of by $P(B)$. The trap answer is the numerator alone (e.g., 0.005 in Example 1) without the normalization.

Bottom line

• Event space is the full list of mutually exclusive outcomes; events are subsets. Probabilities sum to 1 across the event space.
• Overlapping events: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$; add directly only when mutually exclusive. If one event is a subset of the other, the union equals the larger probability.
• Independent means $P(A\cap B)=P(A)P(B)$; mutually exclusive means $P(A\cap B)=0$. They are not the same: mutually exclusive events with positive probabilities CANNOT be independent.
• Conditional independence is independence given a third variable; defaults are usually conditionally independent given a common factor like the macro state, the linchpin of factor-based credit models.

Exam shortcut

When a question gives you a conditional probability and asks for the reverse direction, it's a Bayes problem: write down prior, likelihood, and the law-of-total-probability marginal, then plug in. Never guess; the trap answers are always the unnormalized numerator or the wrong denominator. Memory aid: "Mutually exclusive means together = zero; independent means knowing-doesn't-help." And for transformations: "Mean stretches and shifts; variance squares the stretch."

The full lesson (about 2,807 words, 19 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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