Bayes' Formula

Builds onTotal Probability Rule — if this page feels steep, start there.

P(A\mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}

$P(A \mid B)$: read right-to-left: the probability of A, GIVEN that B has been observed
$P(B \mid A)$: the reverse conditional: how likely the evidence B is when A is true
$P(A)$: the prior: what you believed about A before any evidence arrived
$P(B)$: how likely the evidence is overall — dividing by it rescales the update onto a 0-to-1 scale

Why it must be true

Bayes' formula is just updating a belief with evidence. You start with a prior $P(A)$ ; evidence $B$ arrives; the new belief $P(A\mid B)$ weights the prior by how strongly the evidence points to $A$ — the ratio $P(B\mid A)/P(B)$ .

The classic exam trap is confusing $P(A\mid B)$ with $P(B\mid A)$ : "the probability a company that beat earnings was upgraded" is not "the probability an upgraded company beats earnings". Bayes is the machine that converts one into the other.

The derivation

Both conditional probabilities describe the same joint event $A \cap B$ :

P(A \cap B) = P(A\mid B)\,P(B) = P(B\mid A)\,P(A)

Set the two expressions equal and divide by $P(B)$ :

P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}

When to reach for it

A probability must be UPDATED after evidence arrives — a downgrade, a positive test, a signal — especially when you know the reverse conditional.

Listen for

“given that … has occurred”“revise / update the probability”“the test came back positive”“after the downgrade / announcement”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Count 100 cases instead of juggling formulas: 20 defaults of which 90% get downgraded (18), and 80 non-defaults... build the little 2×2 table, then read the answer as a simple fraction.
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The posterior lies between the prior and certainty, dragged by the evidence's strength. If your 'updated' probability moved the wrong direction, you've swapped the conditionals.

Traps in applying it

✗Swapping P(A|B) for P(B|A) — the prosecutor's fallacy, and the most-tested trap.
✗Neglecting the base rate: strong evidence about a rare event still leaves a small posterior.
✗Stopping at the joint probability P(A and B) without dividing by P(B).

Limits & criticisms

Bayes is only as good as its prior, which is often subjective — the historic frequentist objection. The likelihood model P(B|A) can be misspecified, and in sequential use, errors compound update after update. None of that dents the identity itself; it disciplines the inputs you feed it.

Where it came from

Rev. Thomas Bayes never published his theorem; his friend Richard Price presented it posthumously in 1763. Laplace rediscovered and generalized it, using it for everything from astronomy to court testimony. It then spent much of the 20th century out of fashion before roaring back as computing made Bayesian updating practical.

Today it updates default probabilities after a downgrade, filters spam, powers medical screening logic, and underlies the machine-learning systems reshaping finance — all with the same one-line update Bayes wrote about drawing billiard balls.

Drill this formula Back to the library