Analyst Dojo

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 4 of Quantitative Methods 6–9% of the exam

Probability Trees and Conditional Expectations

3 lessons. About 20 minutes in cram mode, 90 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

  1. 1
  2. 2
    Probability trees and conditional expected values

    calculate 8 min cram, 32 min full

  3. 3

What the exam expects you to do

  1. Calculate expected values, variances, and standard deviations and demonstrate their application to investment problems.Expected value, variance and standard deviation of a forecast
  2. Formulate an investment problem as a probability tree and explain the use of conditional expectations in investment application.Probability trees and conditional expected values
  3. Calculate and interpret an updated probability in an investment setting using Bayes’ formula.Bayes' formula: updating a probability with new information

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

How deeply to learn each part
Know the idea
Understand it in general. It frames other topics, but you will not calculate with it.
Use it
Understand it well enough to solve a question with it. Recognize when it applies.
Memorize
Learn it by heart. The exam will not give it to you.
1

Expected value, variance and standard deviation of a forecast

Known

MemorizeForecast formulas

MeasureFormulaWeights
Expected valueΣ P(x) × xprobabilities
VarianceΣ P(x) × (x − E)²probabilities
Std. deviation√variancesame units as x

Memory hook Same shapes as the historical formulas in LM3, with the weight 1/n (or 1/(n − 1)) swapped for the probability P.

Know it cold Forecast with probabilities: weight by P, never divide by n − 1. History: sample formulas with n − 1.

Key terms

  • probability distribution Know the idea
  • scenario analysis Know the idea
  • expected value of a random variable Memorize
  • variance of a random variable Memorize
  • standard deviation Memorize
From a table of scenarios to risk and reward
Probability distribution of outcomeseach outcome with its probability; probabilities add to 1
  • Expected value Memorize Σ P × x, the probability-weighted average
    • Reads as the forecast, not a promised outcome
  • Variance Memorize Σ P × (x − E)², units are squared
    • Weights probabilities, so no n − 1
  • Standard deviation Memorize √variance, same units as the outcomes
    • Reads as the typical distance from the forecast

Formulas

Expected value Memorize
E(X)=P(x1) x1+P(x2) x2+⋯+P(xn) xn=∑i=1nP(xi) xiE(X) = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i
Quick example

A return is 10% with probability 0.4 and 4% with probability 0.6. E = 0.4 × 10 + 0.6 × 4 = 6.4%.

On your calculator
  1. 0.4×10+(0.6×4)= 6.4
  1. 0.4ENTER10×0.6ENTER4×+ 6.4

ENTER enter (on the HP, puts the number on the stack)

Variance of a forecast Memorize
σ2(X)=∑i=1nP(xi) [xi−E(X)]2\sigma^2(X) = \sum_{i=1}^{n} P(x_i)\,[x_i - E(X)]^2
Quick example

Same forecast, E = 6.4. 0.4 × (10 − 6.4)² + 0.6 × (4 − 6.4)² = 5.184 + 3.456 = 8.64 (%²).

On your calculator
  1. (10−6.4)x²×0.4+((4−6.4)x²×0.6)= 8.64

x² square

  1. 10ENTER6.4−ENTER×0.4×4ENTER6.4−ENTER×0.6×+ 8.64

ENTER enter (on the HP, puts the number on the stack)

Standard deviation of a forecast Memorize
σ(X)=σ2(X)\sigma(X) = \sqrt{\sigma^2(X)}
Quick example

Variance 8.64 (%²). √8.64 = 2.94%.

On your calculator
  1. 8.64√x 2.94

√x square root

  1. 8.64g√x 2.94

g blue shift key · √x square root

2

Probability trees and conditional expected values

Known

MemorizeTree rules

ItemRule
Branches from one nodeprobabilities add to 1
End-branch (joint) probabilityproduct along the path: P(S) × P(x | S)
All end branchesadd to 1
Conditional E(X | S)weight outcomes by P(x | S) inside branch S
Unconditional E(X)Σ E(X | Sᵢ) × P(Sᵢ)

Memory hook Fold the tree from the right: each first-stage node becomes one number, then weight those numbers by the first-stage probabilities.

Know it cold Total probability rule needs scenarios that are mutually exclusive and exhaustive, not equally likely.

Key terms

  • probability tree diagram Use it
  • node Know the idea
  • conditional probability Use it
  • conditional expected value Memorize
  • total probability rule for expected value Memorize
  • conditional variance Know the idea
Reading a two-stage probability tree
Probability treethe first uncertainty, then the outcomes within each branch
  • Node Know the idea a point where uncertainty resolves
    • Branch probabilities from one node they add to 1
  • End-branch (joint) probability Use it multiply along the path
    • All end branches add to 1
  • Conditional expected value Memorize the forecast within one first-stage branch
    • Unconditional E(X) weight each by its branch probability and add

Formulas

Joint probability along a path Use it
P(A and B)=P(B∣A) P(A)P(A \text{ and } B) = P(B \mid A)\,P(A)
Quick example

A scenario has probability 0.30, and an outcome within it has conditional probability 0.80. The joint probability is 0.30 × 0.80 = 0.24.

On your calculator
  1. 0.3×0.8= 0.24
  1. 0.3ENTER0.8× 0.24

ENTER enter (on the HP, puts the number on the stack)

Conditional probability Use it
P(A∣B)=P(A and B)P(B),P(B)≠0P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}, \quad P(B) \neq 0
Quick example

Joint probability 0.24, P(B) = 0.30. P(A | B) = 0.24 ÷ 0.30 = 0.80.

On your calculator
  1. 0.24÷0.3= 0.8
  1. 0.24ENTER0.3÷ 0.8

ENTER enter (on the HP, puts the number on the stack)

Conditional expected value Memorize
E(X∣S)=P(x1∣S) x1+P(x2∣S) x2+⋯+P(xn∣S) xnE(X \mid S) = P(x_1 \mid S)\,x_1 + P(x_2 \mid S)\,x_2 + \cdots + P(x_n \mid S)\,x_n
Quick example

Given scenario S, an outcome of 12 has probability 0.8 and an outcome of 4 has 0.2. E(X | S) = 9.6 + 0.8 = 10.4.

On your calculator
  1. 0.8×12+(0.2×4)= 10.4
  1. 0.8ENTER12×0.2ENTER4×+ 10.4

ENTER enter (on the HP, puts the number on the stack)

Total probability rule for expected value Memorize
E(X)=E(X∣S1) P(S1)+E(X∣S2) P(S2)+⋯+E(X∣Sn) P(Sn)E(X) = E(X \mid S_1)\,P(S_1) + E(X \mid S_2)\,P(S_2) + \cdots + E(X \mid S_n)\,P(S_n)
Quick example

E(X | S₁) = 3.0 with P(S₁) = 0.7, and E(X | S₂) = −1.0 with P(S₂) = 0.3. E(X) = 2.1 − 0.3 = 1.8.

On your calculator
  1. 0.7×3+(0.3×1+|-)= 1.8

+|- change sign

  1. 0.7ENTER3×0.3ENTER1CHS×+ 1.8

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Conditional variance Know the idea
σ2(X∣S)=∑iP(xi∣S) [xi−E(X∣S)]2\sigma^2(X \mid S) = \sum_{i} P(x_i \mid S)\,[x_i - E(X \mid S)]^2
Quick example

Given S, outcomes 5 and −1 each have probability 0.5, so E(X | S) = 2. Variance = 0.5 × 9 + 0.5 × 9 = 9.

On your calculator
  1. (5−2)x²×0.5+((1+|-−2)x²×0.5)= 9

x² square · +|- change sign

  1. 5ENTER2−ENTER×0.5×1CHSENTER2−ENTER×0.5×+ 9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

3

Bayes' formula: updating a probability with new information

Known

MemorizeBayes' update

PieceSymbolMeaning
PriorP(E)before the news
LikelihoodP(I | E)news if the event is true
JointP(I | E) × P(E)news and event
InformationP(I)sum of every joint for the news
PosteriorP(E | I)joint ÷ P(I)

Memory hook The posterior is your share of all the ways the news could have happened: the event branch over every branch with the news.

Know it cold Posterior = likelihood × prior ÷ P(I). Posteriors over all scenarios add to 1.

MemorizeDiffuse prior

Memory hook No reason to favor any scenario, so give each the same prior. The priors then cancel and the likelihoods alone set the posterior.

Know it cold Equal priors: posterior = likelihood ÷ sum of the likelihoods.

Key terms

  • prior probability Memorize
  • likelihood Memorize
  • posterior probability Memorize
  • Bayes' formula Memorize
  • joint probability Use it
  • total probability rule Use it
  • diffuse prior Use it
The four pieces of a Bayes update
Updating a probabilitynews arrives, so revise the view
  • Prior P(E) Memorize your probability before the news
  • Likelihood P(I | E) Memorize how likely the news is if the event is true
  • P(I) Memorize total chance of the news: add every branch that produces it
    • Event branch P(I | E) × P(E)
    • No-event branch P(I | not E) × P(not E)
  • Posterior P(E | I) Memorize event branch ÷ P(I)

Formulas

Bayes' formula Memorize
P(E∣I)=P(I∣E)P(I) P(E)P(E \mid I) = \dfrac{P(I \mid E)}{P(I)}\,P(E)
Quick example

P(E) = 0.20, P(I | E) = 0.90, P(I) = 0.30. P(E | I) = 0.90 × 0.20 ÷ 0.30 = 0.60.

On your calculator
  1. 0.9×0.2=÷0.3= 0.6
  1. 0.9ENTER0.2×0.3÷ 0.6

ENTER enter (on the HP, puts the number on the stack)

Probability of the information Memorize
P(I)=P(I∣E) P(E)+P(I∣Ec) P(Ec)P(I) = P(I \mid E)\,P(E) + P(I \mid E^c)\,P(E^c)
Quick example

P(E) = 0.20, P(I | E) = 0.90, P(I | not E) = 0.10. P(I) = 0.18 + 0.08 = 0.26.

On your calculator
  1. 0.2×0.9+(0.8×0.1)= 0.26
  1. 0.2ENTER0.9×0.8ENTER0.1×+ 0.26

ENTER enter (on the HP, puts the number on the stack)

Bayes' formula with several scenarios Use it
P(Si∣I)=P(I∣Si) P(Si)∑jP(I∣Sj) P(Sj)P(S_i \mid I) = \dfrac{P(I \mid S_i)\,P(S_i)}{\sum_{j} P(I \mid S_j)\,P(S_j)}
Quick example

Priors 0.6, 0.3, 0.1 and likelihoods 0.2, 0.5, 0.9. P(I) = 0.12 + 0.15 + 0.09 = 0.36. Posterior of the first: 0.12 ÷ 0.36 = 0.333.

On your calculator
  1. 0.6×0.2=STO1
  2. +(0.3×0.5)+(0.1×0.9)= 0.36
  3. 1/x×RCL1= 0.3333

STO store · 1/x reciprocal · RCL recall a stored number

  1. 0.6ENTER0.2×STO1
  2. 0.3ENTER0.5×+0.1ENTER0.9×+ 0.36
  3. 1/xRCL1× 0.3333

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

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