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Quantitative MethodsModule 4 of 11

Probability Trees and Conditional Expectations

6

Concepts

4

Formulas

1

Decisions

4

Quiz Questions

Key Concepts

6 concepts covered in this module.

Expected Value

E(X) = Σ P(xi) × xi. The probability-weighted average of all possible outcomes.

Conditional Probability

P(A|B) = P(AB) / P(B). The probability of A given that B has occurred.

Total Probability Rule

P(A) = P(A|B)P(B) + P(A|Bc)P(Bc). Weights conditional probabilities by scenario probabilities.

Bayes' Formula

Updates prior probabilities with new information: P(B|A) = P(A|B) × P(B) / P(A). Crucial for updating investment views.

Probability Trees

Visual tool showing sequential conditional probabilities. Multiply along branches, add across endpoints.

Variance via Expectations

Var(X) = E(X²) - [E(X)]² = Σ P(xi)(xi - E(X))²

Formulas

4 essential formulas for this module.

Expected Value

E(X) = Σ P(xi) × xi

Where: P(xi) = probability of outcome i

Bayes' Formula

P(B|A) = [P(A|B) × P(B)] / P(A)

Where: P(B) = prior, P(B|A) = posterior

Total Probability Rule

E(X) = E(X|S1)P(S1) + E(X|S2)P(S2) + ...

Where: Si = mutually exclusive scenarios

Multiplication Rule

P(AB) = P(A|B) × P(B)

Where: Joint probability of A and B

Decision Frameworks

1 decision frameworks to guide your analysis.

When to use Bayes' Formula?

  • Updating probability estimates with new evidence
  • Medical/diagnostic testing problems
  • Revising economic forecasts with new data

Mind Map

Visual overview of how concepts connect in this module.

Probability & Expectations
Expected Value
E(X) = Σ P(x)x
Variance = E(X²) - [E(X)]²
Conditional expectation
Probability Rules
Addition rule
Multiplication rule
Total probability rule
Complement rule
Bayes' Formula
Prior probability
Likelihood
Posterior probability
Evidence (normalizing)
Probability Trees
Sequential events
Multiply along branches
Add across endpoints
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Expected Value

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Answer
E(X) = Σ P(xi) × xi. The probability-weighted average of all possible outcomes.
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