Probability and sampling - Conditional probability and Bayes...

Learning Outcomes

This article explains how to work confidently with conditional probability and Bayes’ theorem in a CFA Level 1 context. It clarifies the difference between marginal, joint, and conditional probabilities, and shows how these measures relate through the multiplication rule and the total probability rule. You learn how to construct, label, and interpret probability trees to represent sequential or scenario-based investment problems, and how to read joint and conditional probabilities directly from the tree. The article also explains how to calculate unconditional default, upgrade, or outperformance probabilities by combining scenario probabilities with conditional outcomes. It demonstrates how Bayes’ theorem converts prior beliefs and likelihood evidence into posterior probabilities, and how this process supports credit analysis, manager selection, and other investment decisions. Throughout, the focus is on setting up problems carefully, identifying the correct conditional terms, and checking that all probabilities reconcile to one, so that you can avoid common exam errors and present clear, numerically justified probability statements.

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are expected to understand the principles and application of conditional probability and Bayes’ theorem within the context of probability and investment problems, with a focus on the following syllabus points:

• Calculating and interpreting conditional probabilities and joint probabilities
• Constructing and analyzing probability trees to represent investment scenarios
• Applying the total probability rule for unconditional probabilities and expected value
• Updating probabilities using Bayes’ theorem when new information becomes available
• Using these techniques in real investment problem settings and communicating your reasoning clearly

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. If event A and event B are not independent, how would you calculate the probability of A and B occurring together (joint probability)?
2. In the context of an investment decision, what does a conditional probability represent?
3. State Bayes’ theorem in symbols and explain what each part means.
4. How can probability trees assist you in visualizing complex scenarios in investment analysis?

Introduction

Probability is a core area in financial analysis, supporting risk estimation, forecasting, and every step of investment decision-making. Many investment problems involve events that are related or occur in sequence, requiring an understanding of conditional probability: the chance of one event given another has happened. Conditional probability and Bayes’ theorem allow you to update beliefs when new information becomes available, an essential skill for the CFA exam.

Key Term: conditional probability
The probability of an event occurring given that another event has already occurred. Denoted $P(A\,|\,B)$, read as "the probability of A given B."

Key Term: joint probability
The probability that two events both occur together. Denoted $P(A \cap B)$ or $P(AB)$.

Key Term: marginal probability
The unconditional probability of an event occurring, regardless of other events. Denoted $P(A)$.

Key Term: Bayes’ theorem
A formula for updating the probability of a hypothesis or event in light of new information.

Conditional Probability and Probability Trees

Conditional probability deals with calculating the likelihood of an event, given that another related event has occurred. For example, an analyst may want to know $P(\text{default} \mid \text{recession})$: the probability that a bond defaults, given there is a recession.

The general definition is:

where $P(A \cap B)$ is the probability that both events occur, and $P(B)$ is the probability of event B.

Probability trees visually represent sequential or conditional events. Each branch splits according to the chance of the next event, given the earlier outcomes. Trees help organize calculations for joint, conditional, and marginal probabilities.

Worked Example 1.1

A credit analyst estimates the probability the market is strong is 0.4, and weak is 0.6. If the market is strong, the probability that a firm’s bond defaults is 0.02; if weak, the default probability is 0.12. What is the overall probability the bond defaults?

Answer:

• Draw a probability tree: split first by strong (0.4) and weak (0.6), and then for each, branch into default/no-default.
• $P(\text{default}) = P(\text{strong}) \times P(\text{default}|\text{strong}) + P(\text{weak}) \times P(\text{default}|\text{weak})$
• $= 0.4 \times 0.02 + 0.6 \times 0.12 = 0.008 + 0.072 = 0.08$
• So the probability the bond defaults is 8%.

Total Probability Rule and Joint Probabilities

The joint probability $P(A \cap B)$ is interpreted as the chance that both A and B happen together. The multiplication rule for probabilities is:

or equivalently,

When you have a set of mutually exclusive scenarios $S_i$ (e.g., economic regimes) that add up to 1 (collectively exhaustive), the total probability of an event $A$ is:

Key Term: total probability rule
States that the overall (marginal) probability of an event is the weighted average of its conditional probabilities across all possible, mutually exclusive scenarios.

Bayes’ Theorem: Probability Updates

Bayes’ theorem is used when you want to revise previously estimated probabilities based on new evidence. It relates conditional and marginal probabilities as follows:

where $H$ is your hypothesis or prior event (e.g., economic strength), and $E$ is new evidence (e.g., an analyst signal).

• $P(H)$ = prior probability of H ("before" seeing E)
• $P(E\,|\,H)$ = probability of observing E, given H
• $P(E)$ = probability of observing E under all scenarios (sum over all possible H)
• $P(H\,|\,E)$ = posterior (updated) probability of H after E is observed

Worked Example 1.2

An investment firm estimates the probability that a manager is skilled (S) as 0.3. Skilled managers have a 70% chance of beating the benchmark; unskilled managers, 20%. If the manager beats the benchmark, what is the updated probability the manager is skilled?

Answer:

• Prior: $P(S) = 0.3$, $P(\text{not S}) = 0.7$
• $P(\text{beat} | S) = 0.7$, $P(\text{beat}|\text{not S}) = 0.2$
• Use total probability to get $P(\text{beat}) = 0.3 \times 0.7 + 0.7 \times 0.2 = 0.21 + 0.14 = 0.35$
• Then, Bayes’ formula:
• $P(S|\text{beat}) = \dfrac{0.7 \times 0.3}{0.35} = 0.6$
• So, after seeing the outperformance, the chance the manager is skilled rises from 0.3 to 0.6.

Exam Warning

Bayes’ theorem and conditional probability tree calculations are common sources of mistakes in the CFA exam. Carefully distinguish prior, conditional, and updated (posterior) probabilities, and check that all probabilities add up correctly.

Summary

Conditional probability and Bayes’ theorem provide a basis for rigorous probability reasoning. You can calculate the chance of complex events by breaking them into conditional or joint probabilities, use probability trees for clarity, and update beliefs using Bayes’ formula. These skills are directly tested and widely applied within investment analysis in the CFA exam.

Key Point Checklist

This article has covered the following key knowledge points:

• Calculate conditional, joint, and marginal probabilities in investment problems
• Apply the total probability rule for marginal probabilities and expected value
• Draw and analyze probability trees to clarify complex scenarios
• Use Bayes’ theorem to revise probabilities in light of new information
• Distinguish between prior, posterior, and likelihood probabilities in updates
• Avoid common mistakes with Bayesian updating and conditional trees

Key Terms and Concepts

• conditional probability
• joint probability
• marginal probability
• Bayes’ theorem
• total probability rule