🎲 Probability — Clear Notes (with Worked Example)
All main formulas + step-by-step example using your Sales & Dividend numbers (56, 8, 24, 12).
1. Definition & Basic Rules
- Probability of event E:
P(E) = (favourable outcomes) / (total outcomes) - Range:
0 ≤ P(E) ≤ 1 - Normalization: Sum of probabilities over the sample space =
1 - Impossible:
P(E)=0· Certain:P(E)=1
2. Types of Events
- Exhaustive: Events that cover the entire sample space (ΣP=1).
- Mutually exclusive: Cannot happen together →
P(A ∩ B)=0. - Not mutually exclusive: May overlap →
P(A ∩ B)>0. - Independent: One does not affect the other →
P(A ∩ B)=P(A)P(B). - Dependent: One affects the other →
P(A ∩ B)=P(A)P(B|A).
3. Key Formulas — Short List
Addition (Union): P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication (Joint): Independent: P(A ∩ B)=P(A)P(B); Dependent: P(A ∩ B)=P(A)P(B|A)
Conditional: P(A|B)=P(A ∩ B)/P(B)
Total probability (two-branch): P(A)=P(B)P(A|B)+P(Bᶜ)P(A|Bᶜ)
Bayes' theorem: P(B|A)= [P(A|B)P(B)] / [P(A|B)P(B)+P(A|Bᶜ)P(Bᶜ)]
Odds: In favour = P/(1−P) ; Against = (1−P)/P
4. Worked Example — Sales (A) & Dividend (B)
Use the grid from your notes (counts are out of 100):
Interpreting as probabilities (divide counts by 100):
P(A) = P(Sales High) = 80/100 = 0.80P(Aᶜ) = 20/100 = 0.20P(B|A) = P(Dividend High | Sales High) = 56/80 = 0.70P(B|Aᶜ) = P(Dividend High | Sales Low) = 8/20 = 0.40
Joint probabilities (direct from grid)
P(A ∩ B) = 56/100 = 0.56(Sales High & Dividend High)P(A ∩ Bᶜ) = 24/100 = 0.24(Sales High & Dividend Low)P(Aᶜ ∩ B) = 8/100 = 0.08(Sales Low & Dividend High)P(Aᶜ ∩ Bᶜ) = 12/100 = 0.12(Sales Low & Dividend Low)
Total Probability — compute P(B) from branches
P(B) = P(A)·P(B|A) + P(Aᶜ)·P(B|Aᶜ)
P(B) = 0.80×0.70 + 0.20×0.40 = 0.56 + 0.08 = 0.64 → P(Dividend High) = 64%.
Bayes — reverse conditioning: P(A|B)
P(A|B) = [P(B|A)·P(A)] / P(B)
0.70×0.80 = 0.56denominator =
P(B) = 0.64⇒
P(A|B) = 0.56 / 0.64 = 0.875 → 87.50%.
Check: Conditional formula consistency
We already have P(A ∩ B)=0.56 and P(B)=0.64. By definition P(A|B)=P(A∩B)/P(B)=0.56/0.64 → same 0.875. (Always good to cross-check.)
Odds — quick
Odds in favour of B (Dividend High): P(B)/(1−P(B)) = 0.64 / 0.36 = 1.777... ≈ 1.78 : 1
Odds against B: 0.36 / 0.64 = 0.5625 ≈ 9 : 16 (if scaled)
5. Quick revision tips
- Think in branches: split by A / Aᶜ, multiply along each branch, add across branches to get a marginal probability.
- Joint = probability inside a cell (branch multiplier). Conditional reverses joint & marginal (
P(A|B)=P(A∩B)/P(B)). - Bayes is just “take the branch you care about / divide by total across branches”.
6. Summary Table (Formulas)
| Concept | Formula | Short use |
|---|---|---|
| Conditional | P(A|B)=P(A∩B)/P(B) |
Prob. given info |
| Joint | P(A∩B)=P(A)P(B|A) |
Both occur |
| Total probability | P(B)=∑ P(branch)·P(B|branch) |
Marginal from branches |
| Bayes | P(A|B)=P(B|A)P(A)/[Σ P(B|branch)P(branch)] |
Reverse conditioning |
If you want, I can now (1) add a 4-question interactive quiz using these numbers, or (2) create a printable one-page PDF of this note. Which one shall I make next?
Vertical Probability Tree — Sales (A) → Dividend (B)
| Sales High (A) | Sales Low (Aᶜ) | Total | |
|---|---|---|---|
| Dividend High (B) | 56 | 8 | 64 |
| Dividend Low (Bᶜ) | 24 | 12 | 36 |
| Total | 80 | 20 | 100 |
- Unconditional (marginal): P(A)=80/100 =
0.80; P(Aᶜ)=20/100 =0.20. - Conditional: P(B|A) = number in (A∩B) ÷ total in A = 56 ÷ 80 =
0.70.
P(B|Aᶜ) = 8 ÷ 20 =0.40. - Joint: P(A ∩ B) = P(A) × P(B|A) = 0.80 × 0.70 =
0.56(or 56/100). Similarly compute the other three joints. - Marginal P(B) (Total probability): add branches that end in B:
P(B) = P(A ∩ B) + P(Aᶜ ∩ B) = 0.56 + 0.08 =0.64. - Bayes (Reverse conditioning): P(A|B) = P(A ∩ B) / P(B). Substitute:
P(A|B) = 0.56 ÷ 0.64 = 56/64 = 7/8 =0.875 = 87.5%.
- What Bayes tells you here: If you observe a High Dividend (B), it's very likely sales were High — P(A|B)=87.5%. This is 'updating belief' after seeing evidence.
- Why total probability matters: It converts branch-level (conditional) knowledge into an overall probability for the event (P(B)).
- Useful memory trick: "Multiply down branches, add across branches." Multiply unconditional × conditional to get each leaf (joint). Add joint leaves for totals.
- Common FRM use: Bayes is used widely in credit scoring, fraud detection and model updating — you observe an outcome and update the probability of the cause.
P(A∩B) = P(A)·P(B|A)(joint)P(A|B) = P(A∩B) / P(B)(conditional via Bayes)P(B) = Σ branches P(branch)·P(B|branch)(total probability)Odds in favour = P/(1−P)(if you need odds)