Start from the definition of conditional probability written two ways for the same joint event: P(A ∩ B) = P(A | B)·P(B), and equally P(A ∩ B) = P(B | A)·P(A). Set those equal and solve for P(A | B), and you have derived (not merely stated) Bayes' theorem:
P(A | B) = P(B | A) · P(A) / P(B)In finance, A is almost always "the thing you actually care about but cannot observe directly" (a company is truly manipulating earnings; a bond will actually default; a manager's outperformance is genuine skill) and B is "the evidence you can observe" (a screen flags the filing; a downgrade warning fires; three good years in a row). Call P(A) the prior (your belief before the evidence) and P(A | B) the posterior, your updated belief after it. The denominator, P(B), is computed by the total probability rule, splitting on whether A is true or false:
P(B) = P(B | A)·P(A) + P(B | not A)·P(not A)Advertisement
Worked example: a screen for a rare condition
Finance and screening language gives P(B | A) and P(B | not A) their own names: P(B | A) is the screen's sensitivity (true-positive rate, how often it correctly flags a real case), and P(B | not A) is its false-positive rate (1 minus its specificity, how often it wrongly flags a clean one). A quick, deliberately generic illustration to fix the mechanism before the full financial version in Worked Example 1: suppose a screening tool for any rare condition has 90% sensitivity and a 5% false-positive rate, and the condition's true prevalence (the prior) is only 1%. Then P(flag) = 0.90×0.01 + 0.05×0.99 = 0.009 + 0.0495 = 0.0585, and P(condition | flag) = 0.009/0.0585 ≈ 15.4%, a screen that sounds excellent (90% sensitive!) still leaves the flagged case more likely not to have the condition than to have it, purely because the condition is rare. This is base-rate neglect: the single most consequential Bayes' theorem mistake in finance, and the reason M2.07 was careful to say, of its own screens, that "screens point; filings convict." This module now makes that warning arithmetic. Worked Example 1 runs the identical logic on a fraud-screen scenario with full numbers, and a useful cross-check exists: the odds form of Bayes' theorem, posterior odds = prior odds × likelihood ratio (where the likelihood ratio is sensitivity divided by the false-positive rate), gives you a second, independent path to the identical posterior, a built-in verification habit worth carrying into every Bayes calculation you ever do.
{
"skill": "Turning a screen's sensitivity, false-positive rate, and base rate into a posterior probability without falling for base-rate neglect",
"use": "Have a chat assistant set up and solve a Bayes'-theorem problem you describe in plain English (a red-flag screen, a diagnostic model, a covenant-warning system) so you can check your own by-hand setup against an independent one.",
"tools": ["Chat assistants — Claude, ChatGPT, Gemini", "A spreadsheet or a scratch calculator for the arithmetic itself"],
"prompt": "I have a screen with sensitivity <S>, false-positive rate <F>, and a base rate of <B> in the population I'm applying it to. Walk me through Bayes' theorem step by step to find P(true condition | flagged), showing the total-probability denominator explicitly, then re-derive the same answer using the odds form (prior odds × likelihood ratio) as an independent cross-check.",
"verify": "This is exactly the calculation most likely to be silently flipped — P(flag | condition) swapped for P(condition | flag), the base-rate fallacy in its purest form — by fluent-sounding, wrong prose. Recompute the total-probability denominator yourself, confirm the odds-form answer matches the direct-form answer to the decimal, and trace the sensitivity/false-positive-rate/base-rate inputs themselves to a named source before any of this enters real work (Guardrail 1–5, especially #3 Recompute).",
"diy": "You must be able to state Bayes' theorem from the definition of conditional probability yourself first — that is exactly what this section just built. An AI can set up the algebra faster than you today; it cannot install the base-rate reflex that keeps you from trusting a screen more than its own math supports.",
"market": "IN"
}