In plain English
Probability is bookkeeping for uncertainty. It keeps your beliefs consistent so that the numbers you quote cannot contradict each other.
The advanced view
The axioms (non-negativity, normalisation, additivity over disjoint events) generate everything else. Conditional probability defines dependence, expectation is a linear operator regardless of dependence, and variance is not — which is exactly why covariance appears in portfolio mathematics.
Probability is the language of uncertainty, and finance is the business of pricing uncertainty. A random variable maps outcomes to numbers: discrete variables have probability mass functions, continuous ones have densities, and the CDF gives P(X ≤ x) for both. Bayes' theorem updates probabilities with new evidence — it is how credit models revise default probabilities, how fraud detection works, and how diagnostic tests are interpreted.
P(A|B) = P(B|A) × P(A) / P(B) E(X) = Σ p_i × x_i Var(X) = E[(X − μ)²] = E(X²) − μ² P(A∪B) = P(A) + P(B) − P(A∩B)
Key distributions: normal (returns over short intervals), log-normal (prices, because returns are roughly normal and prices cannot go negative), binomial (successes in n trials, and the basis of binomial option pricing), Poisson (rare-event counts such as defaults or claims) and Student's t (heavier tails, for small samples and fat-tailed returns). The Central Limit Theorem says sample means converge to normal regardless of the underlying distribution; the Law of Large Numbers says they converge to the population mean. CLT tells you the shape, LLN the target.
Essential vocabulary
- Expected value E(X)
- Probability-weighted average of outcomes. The long-run average.
- Variance Var(X)
- Expected squared deviation from the mean. Standard deviation is its root, in the units of X.
- Covariance & correlation
- Covariance measures co-movement; correlation normalises it to [−1, 1]. Critical for portfolio construction.
- Conditional probability
- P(A|B): probability of A given B has occurred. Changes the sample space.
- Independence
- P(A∩B) = P(A)×P(B). Most financial assets are not independent.
Why it works
Bayes' theorem works because it is just the definition of conditional probability read both ways: P(A|B)P(B) = P(A∩B) = P(B|A)P(A). It corrects for base rates because the denominator counts every way the evidence could have arisen, not only the way you had in mind — the reason rare-condition test results are so often misread.
Common pitfalls
- ×Ignoring the base rate when a test is accurate but the condition is rare.
- ×Adding probabilities of events that are not mutually exclusive.
- ×Assuming independence to make the arithmetic easy when the case implies correlation.
How it is used — a screening test in your head
Step 1 of 4
- 1Prevalence 1%, sensitivity 99%, false positive rate 5%.
Deeper
Deeper: Bayes, base rates and conditional thinking
P(A|B) = P(B|A) × P(A) ÷ P(B). The prior P(A) — the base rate — is what people drop, which is why a 99%-accurate test for a 1-in-1,000 condition still produces mostly false positives. In business the same trap appears in fraud screening, churn prediction and due-diligence red flags.
Expected value alone is not a decision rule when outcomes are lumpy: the variance and the possibility of ruin matter. That is why firms buy insurance with a negative expected value, and why 'positive EV' bets with a 5% chance of bankruptcy are declined.
Must know cold
- ✓P(A|B) = P(B|A)P(A) ÷ P(B).
- ✓Independent events multiply; mutually exclusive events add.
- ✓E[X] = Σ p·x; Var(X) = E[X²] − (E[X])².
- ✓Base rates dominate when the event is rare.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
A fraud model flags 1% of clean transactions and catches 95% of fraud. Fraud is 0.2% of volume. A transaction is flagged — how likely is fraud?
Exercise 2
A project pays +100 with probability 0.6 and −80 with probability 0.4. Would you run it once? Fifty times?
The critical values you quote are just cut-points on this density: ±1.645 for 90%, ±1.96 for 95%, ±2.58 for 99%. Knowing the shape means you can sanity-check any z-score or p-value in your head.