Probability Foundations for Conditional Expectations

Updated for the 2026-2027 CFA® Level I curriculum.

Probability rules provide the foundation for expected values, probability trees, and Bayesian updating. Before working through those topics, you need to understand how events relate, how probabilities combine, and how new information changes the probability assigned to an outcome.

This study note brings those ideas together in one place, with an emphasis on the rules you will use in later investment problems.

Key Takeaways About the Properties of Probability

• A probability must fall between and .

• The probabilities of all mutually exclusive and exhaustive outcomes must sum to .

• The complement of an event has probability minus the probability of the event.

• Mutually exclusive events cannot occur at the same time.

• Independent events can occur together without changing each other’s probabilities.

• Joint probability measures the chance that two events occur together.

• Conditional probability measures the chance of one event given that another has occurred.

• The addition rule calculates the probability of one event, another event, or both.

• The multiplication rule calculates the probability of events occurring together.

• The total probability rule combines conditional probabilities across all possible states.

What You Need to Know for CFA Level I

For CFA Level I, focus on turning a written investment scenario into the correct probability relationship.

You should be able to:

• Identify an experiment, outcome, sample space, and event.

• Confirm whether probabilities form a valid distribution.

• Calculate the complement of an event.

• Distinguish a union from an intersection.

• Compare mutually exclusive, independent, and dependent events.

• Calculate joint and conditional probabilities.

• Apply the addition and multiplication rules.

• Use the total probability rule across several states.

• Recognize when probabilities are conditional on different information sets.

• Avoid reversing expressions such as and .

What Are the Defining Properties of Probability?

The first property of probability is its range:

A probability of means event is impossible. A probability of means it is certain. Values between and describe different levels of uncertainty.

The second property applies to the full sample space. If the possible outcomes are mutually exclusive and exhaustive, their probabilities must sum to .

Where:

• = possible outcome or state

• = number of possible outcomes

Mutually exclusive means that only one of the outcomes can occur. Exhaustive means that the outcomes cover every possible result.

For example, suppose the economy can be classified as expansion, stable growth, or recession. If those states cannot occur at the same time and no other state is possible, their probabilities must add to .

All probabilities in a calculation must also refer to a consistent sample space and information set. A conditional probability based on an expansion cannot be combined carelessly with an unconditional probability that covers every economic state.

Basic Probability Terms and Event Notation

Probability questions become easier once you can translate the scenario into events.

Union and Intersection Notation

The union of events and is written as:

It includes outcomes where occurs, occurs, or both occur.

The intersection is written as:

It includes only outcomes where both events occur together.

Words such as or usually signal a union. Words such as and, both, or together usually signal an intersection.

Mutually Exclusive, Independent, and Dependent Events

These event relationships answer separate questions.

Mutually Exclusive Events

Mutually exclusive events cannot occur together.

For example, if a bond is classified as either investment grade or high yield under a single rating classification, it cannot belong to both categories at the same time.

Mutually exclusive events with positive probabilities are dependent. Learning that one occurred tells you that the other did not occur.

Independent Events

Two events are independent when the occurrence of one does not change the probability of the other.

Independence can also be tested through the joint probability:

Dependent Events

Events are dependent when information about one event changes the probability of the other.

For example, a company’s chance of beating earnings estimates may be higher during an economic expansion than during a recession. The earnings result and economic state are therefore dependent.

Core Probability Rules

Complement Rule

The complement of event , written as , includes every outcome where does not occur.

If a bond has a 4% probability of default, its probability of avoiding default is:

The probability of no default is 96%.

Addition Rule

The addition rule calculates the probability that event , event , or both events occur.

The intersection is subtracted because outcomes where both events occur are included once in and again in .

When the events are mutually exclusive, their intersection equals zero:

This simplified form should only be used when the events cannot occur together.

Multiplication Rule

The multiplication rule calculates the joint probability that two events occur together.

The same relationship can be written in the opposite order:

Both versions calculate the same intersection. Use the form that matches the conditional probability given in the question.

For independent events, the conditional probability equals the marginal probability:

Conditional Probability Rule

Conditional probability measures the likelihood of event after event is known to have occurred.

This formula requires .

For example, measures the chance of default within the recession state. It does not describe the overall probability of default across every economic state.

Total Probability Rule

The total probability rule calculates the overall probability of an event by considering every possible state in which it can occur.

Where:

• = event being evaluated

• = state

• = probability of event within state

• = probability of state

The states must be mutually exclusive and exhaustive.

For two states, the formula becomes:

This structure also appears in conditional expected value problems. The difference is the quantity being combined. Total probability combines conditional probabilities, while the law of total expectation combines conditional expected values.

How the Probability Rules Connect

Each rule answers a different type of question.

Reading the wording before choosing a formula can save time. Identify whether the question asks about not, or, and, given, or across all states.

Worked Investment Example

Let:

• Event = the economy expands

• Event = a company reports earnings above expectations

Suppose:

The probability of an earnings beat depends on the economic state:

Step 1: Calculate the Complement

The probability that the economy does not expand is:

Step 2: Calculate the Joint Probability of Expansion and an Earnings Beat

Apply the multiplication rule:

There is a 42% probability that the economy expands and the company beats earnings expectations.

Step 3: Calculate the Overall Probability of an Earnings Beat

An earnings beat can occur during an expansion or when the economy does not expand. Apply the total probability rule:

The overall probability of an earnings beat is 54%.

Step 4: Calculate the Probability of an Expansion or an Earnings Beat

Apply the addition rule:

There is a 72% probability that an expansion occurs, an earnings beat occurs, or both occur.

Step 5: Test Whether the Events Are Independent

If the events were independent, the probability of an earnings beat during an expansion would equal the unconditional probability of an earnings beat.

Because these probabilities are different, the events are dependent.

You can confirm the result with the joint-probability test:

The product of the marginal probabilities, , does not equal the actual joint probability of .

The economic state changes the probability of an earnings beat, which is consistent with the scenario.

Common Exam Traps

• Treating mutually exclusive events as independent.

• Adding probabilities without subtracting the intersection.

• Using the simplified addition rule when the events can occur together.

• Multiplying marginal probabilities when the events are dependent.

• Reversing and .

• Confusing a joint probability with a conditional probability.

• Using a conditional probability as though it were the overall probability.

• Forgetting to calculate the complement before applying total probability.

• Applying the total probability rule to states that are not mutually exclusive and exhaustive.

• Combining probabilities that refer to different sample spaces or information sets.

Practice Question

A lender estimates a 30% probability of recession next year.

If a recession occurs, a borrower has a 20% probability of default. If no recession occurs, the borrower has a 5% probability of default.

What is the borrower’s overall probability of default?

1. 6.0%

2. 9.5%

3. 20.0%

FAQs About the Properties of Probability