R.← All explanations03 / PROBABILITY

BAYES’ THEOREM WITHOUT THE FOG

A positive test
is not the answer.

It is evidence. What that evidence means depends on what was likely before the test.

WHAT YOU BELIEVED×NEW EVIDENCE=WHAT TO BELIEVE NOW
01

THE TEST LAB

See all 1,000 people.

True positiveFalse positiveNegative

AFTER A POSITIVE RESULT

16%

actually have the condition.

About 10 true positives sit beside 50 false positives.

10 true positives60 positive tests

Rounded simulation for intuition. Real diagnostic interpretation also depends on population, test thresholds, repeat testing, symptoms, measurement quality, and clinical judgment.

02

THREE LEVERS

The words that change the answer.

PRIOR

Prevalence

Before testing, how common is the condition in this particular population?

“How many start in the red group?”
IF CONDITION

Sensitivity

Among people who really have it, how often does the test correctly say positive?

“How many cases do we catch?”
IF NO CONDITION

Specificity

Among people who do not have it, how often does the test correctly say negative?

“How many healthy people do we clear?”
03

THE TRAP

Accuracy can hide the base rate.

1%

Start rare.

In 1,000 people, only 10 have the condition.

→
95%

Use a good test.

It catches about 10 cases—but a 5% false-positive rate flags about 50 healthy people.

→
~16%

Update honestly.

Only about 1 in 6 positive results is a true case.

THE FORMAL VERSION

P(condition | +)=P(+ | condition) × P(condition)P(+)

You do not need to memorize the symbols. Remember the movement: start with the base rate, weight it by the evidence, then compare it with every way that evidence could appear.