3.3 Probability, Conditional Expectations, and Bayes

CFA Level I Study Guide Exam

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1 Chapter 1: Orientation, Official Sources, and Exam Strategy

1.1 What Level I Tests and Why It Matters 1.2 Official Source Control and 2026 Curriculum 1.3 Eligibility, Registration, Passport, and PSM 1.4 Exam Format, Fees, Windows, and Results 1.5 Topic Weights and Session Map 1.6 Three-Hundred-Hour Study Plan

2 Chapter 2: Ethical and Professional Standards

2.1 Ethics Weight, Code, and Standards Map 2.2 Professionalism and Integrity of Capital Markets 2.3 Duties to Clients and Employers 2.4 Investment Analysis, Recommendations, and Conflicts 2.5 Responsibilities as a CFA Candidate and Member 2.6 GIPS, Performance Presentation, and Ethics Case Lab

3 Chapter 3: Quantitative Methods

3.1 Rates, Returns, and Time Value of Money 3.2 Statistics, Distributions, and Risk Measures 3.3 Probability, Conditional Expectations, and Bayes 3.4 Portfolio Mathematics and Safety-First 3.5 Sampling, Estimation, and Hypothesis Testing 3.6 Regression, Simulation, and Big Data

4 Chapter 4: Economics

4.1 Firm Market Structures and Breakeven Analysis 4.2 Business Cycles and Economic Indicators 4.3 Fiscal Policy, Monetary Policy, and Central Banks 4.4 Geopolitics, Globalization, and Trade 4.5 Foreign Exchange, Capital Flows, and Arbitrage 4.6 Economics Case Lab

5 Chapter 5: Financial Statement Analysis

5.1 FSA Framework and Information Quality 5.2 Income Statements, Revenue, Expense, and EPS 5.3 Balance Sheets, Assets, Liabilities, and Equity 5.4 Cash Flow Statements and Free Cash Flow 5.5 Inventories, Long-Lived Assets, and Taxes 5.6 Reporting Quality, Ratios, DuPont, and Modeling

6 Chapter 6: Corporate Issuers

6.1 Organizational Forms and Corporate Features 6.2 Stakeholders, ESG, and Governance Conflicts 6.3 Working Capital, Liquidity, and Cash Conversion 6.4 Capital Budgeting, NPV, IRR, and Real Options 6.5 Capital Structure, WACC, and Modigliani-Miller 6.6 Business Models and Corporate Case Lab

7 Chapter 7: Equity Investments

7.1 Market Organization, Orders, and Trading 7.2 Security Market Indexes and Efficiency 7.3 Equity Securities and Industry Analysis 7.4 Company Analysis and Competitive Position 7.5 Dividend Discount Models and Valuation Inputs 7.6 Multiples, Private Company Valuation, and Equity Case Lab

8 Chapter 8: Fixed Income

8.1 Bond Features, Issuance, and Cash Flow Structures 8.2 Yield Measures, Spot Rates, and Forward Rates 8.3 Bond Valuation, Clean Price, and Accrued Interest 8.4 Duration, Convexity, and Interest Rate Risk 8.5 Credit Risk, Ratings, and Spreads 8.6 Securitization and Fixed Income Case Lab

9 Chapter 9: Derivatives

9.1 Derivative Markets, Forward Commitments, and Claims 9.2 Forwards, Futures, and Swap Basics 9.3 Option Payoffs, Moneyness, and Risk 9.4 Arbitrage, Pricing, and No-Arbitrage Intuition 9.5 Hedging, Speculation, and Risk Management 9.6 Derivatives Case Lab

10 Chapter 10: Alternative Investments

10.1 Alternatives Role, Characteristics, and Structures 10.2 Hedge Funds, Strategies, Fees, and Risks 10.3 Private Capital, Venture, Buyout, and Private Credit 10.4 Real Estate, Infrastructure, and Commodities 10.5 Liquidity, Due Diligence, and Performance Measurement 10.6 Alternatives Case Lab

11 Chapter 11: Portfolio Management

11.1 Portfolio Return, Risk, and Diversification 11.2 Portfolio Planning, Policy, and Construction 11.3 Risk Management and Investor Objectives 11.4 Behavioral Finance and Client Decision Errors 11.5 Fintech, Data, and Portfolio Applications 11.6 Portfolio Case Lab

12 Chapter 12: Integrated CFA Level I Review

12.1 Session One Integration: Ethics, Quant, Economics, FSA, and Corporate 12.2 Session Two Integration: Equity, Fixed Income, Derivatives, Alternatives, and PM 12.3 Formula Sheet and Calculator Workflow 12.4 Question Stem Reading and Answer Elimination 12.5 Mock Exam Review and Error Log 12.6 Mixed-Topic Case Lab

13 Chapter 13: Final Countdown, Results, and Next Steps

13.1 Ninety-Day, Sixty-Day, and Thirty-Day Plan 13.2 Exam Week Prometric and Passport Checklist 13.3 PSM Completion and Result Release 13.4 If You Pass: Level II Path 13.5 If You Fail: Retake and Deferral Strategy 13.6 Career Path, Charter Requirements, and Membership 13.7 Final Readiness Audit

3.3 Probability, Conditional Expectations, and Bayes

Key Takeaways Explain This Topic AI

Probability rules connect single events, joint events, conditional events, and complements; mutually exclusive and independent are distinct concepts.

Expected value is a probability-weighted average and is the foundation for expected return and variance of a random variable.

The total probability rule combines conditional paths across states into one unconditional probability.

Bayes' formula updates a prior probability after evidence, and base rates strongly influence the posterior.

Last updated: June 2026

Probability, Conditional Expectations, and Bayes

Probability assigns numerical weights to uncertain outcomes. In Level I Quant the application is concrete: expected return, scenario analysis, risk forecasting, credit events, and manager selection all rest on probability rules. Keep events, outcomes, and conditions clearly separated on your scratch paper.

Core probability rules

Probabilities lie between 0 and 1. Empirical probability comes from historical frequency, a priori probability from formal reasoning, and subjective probability from informed judgment. The complement rule is P(A') = 1 - P(A) . The addition rule is P(A or B) = P(A) + P(B) - P(A and B) ; for mutually exclusive events the joint term is zero. Odds for an event are P/(1-P) , and odds against are the reciprocal.

Joint probability is the chance both events occur. Conditional probability is P(A | B) = P(A and B)/P(B) . Rearranging gives the multiplication rule P(A and B) = P(A | B) x P(B) . If A and B are independent , P(A | B) = P(A) , so the joint probability is simply the product P(A) x P(B) . A crucial distinction: mutually exclusive events cannot occur together (joint = 0), whereas independent events can co-occur but neither shifts the other's probability. Two events with positive probabilities cannot be both mutually exclusive and independent.

Expected value, variance, and covariance

Expected value is a probability-weighted average: E(X) = sum p_i x_i . For returns it is expected return. The variance of a random variable is sum p_i [x_i - E(X)]^2 . Two choices can share an expected value yet differ sharply in dispersion, so always pair the expectation with a risk measure. Covariance from scenarios multiplies each asset's deviation from its own expected return, weights the products by state probability, and sums: positive covariance means the assets tend to beat their averages together.

Conditional expectations and total probability

Conditional expected value weights outcomes after a condition is known; expected equity return during an expansion can differ from that during a recession. Analysts build probability trees with economic states in the first branch and asset returns or default outcomes in later branches. The total probability rule combines the branches: with two states P(A) = P(A | B)P(B) + P(A | B')P(B') , and with many states you sum P(A | S_i)P(S_i) over all states. This is the standard route from state-specific default probabilities to an overall default probability.

Bayes' formula

Bayes' formula reverses the conditioning once new evidence arrives. In two-state form, P(B | A) = [P(A | B) x P(B)] / P(A) , where the denominator P(A) usually comes from total probability. The prior is the original probability of B; the posterior is the updated probability after observing the signal A.

• Probability tool Formula Interpretation
• Complement 1 - P(A) Event A does not occur
• Addition P(A) + P(B) - P(A and B) A or B occurs
• Conditional P(A and B)/P(B) A given B
• Multiplication P(A | B) x P(B) A and B both occur
• Independence P(A) x P(B) Joint when independent
• Total probability sum P(A | S_i)P(S_i) Overall chance across states
• Bayes [P(A|B)P(B)]/P(A) Updated state probability
• Counting and exam tactics

Three counting rules appear: the factorial n! orders n items; permutations nPr = n!/(n-r)! choose r ordered from n; and combinations nCr = n!/[(n-r)! r!] choose r when order does not matter. A clean tree prevents most errors: put the first event on the left, conditional events on the right, multiply along branches, and add branch products for unconditional probabilities. For a conditional probability, divide the relevant joint probability by the probability of the condition.

Worked Bayes example

Return to the credit model that flags 70% of future defaulters and 20% of survivors, with a 10% prior default rate. Total probability gives P(flag) = 0.70(0.10) + 0.20(0.90) = 0.07 + 0.18 = 0.25 . The posterior probability that a flagged firm actually defaults is P(default | flag) = P(flag | default) x P(default) / P(flag) = (0.70 x 0.10)/0.25 = 0.07/0.25 = 28% . Even though the model catches 70% of defaulters, a flagged firm has only a 28% chance of defaulting, because survivors vastly outnumber defaulters and contribute most of the flags.

This is the base-rate lesson the exam loves to test: high signal accuracy does not translate into a high posterior when the underlying condition is rare.

Counting refresher and exam tactics

Use the labeling formula n!/(n1! n2! ... nk!) when assigning n items into k labeled groups (for example, rating analysts into buy/hold/sell buckets). Bayes questions reward labeling each probability before plugging in. When a stem describes a test, screen, or signal with a stated accuracy and asks for the probability of the underlying condition, recognize it as a Bayes problem and build the denominator with total probability first. When events are described as independent, never subtract a joint term you have already accounted for, and confirm independence by checking whether P(A and B) equals P(A) x P(B) .

Ignoring base rates leads analysts to overreact to a model alert, a credit warning, or a screening result.