From Conditional Probability to Bayes' Theorem

What “Given” Really Means

A probability is always a share of some crowd. The word “given” doesn't change the question — it changes the crowd. You stop looking at everyone and look only at the ones that match what you already know.

Here are 100 days of weather. Each tile is one day: grey = cloudy, yellow = sunny, and a blue drop means it rained. How likely is rain? It depends on what you already know about the sky.

The Universe Shrinks — From Counting to a Formula

Counting tiles becomes a formula once you swap counts for probabilities. Imagine the whole square has area 1. Then the area of a shape is its probability.

Drag the two rectangles, resize them with their corner dot, then press Given B. The picture zooms until B fills the whole frame. What you see is literally the new universe — and the purple part's share of it is P(A | B).

Notice two things while you play. First, when the rectangles don't overlap, P(A | B) = 0: once you know B happened, A is impossible (they are mutually exclusive). Second, P(A | B) and P(B | A) share the same top — the purple overlap — but divide by different wholes. That is the whole secret of the next section.

Order Matters: P(A | B) ≠ P(B | A)

Same overlap, different crowd. Swapping what is “given” swaps the denominator — and the answer can change enormously.

Independence — When “Given” Changes Nothing

Sometimes zooming in changes nothing: the share of A inside B is the same as the share of A everywhere. Then knowing B tells you nothing about A — the events are independent.

Below, each column is one group of students and the green part shows who passed. Column width is how big the group is. Move the sliders until the two green levels line up — that is what independence looks like.

Probability Trees — The Multiplication Rule

Rearrange the conditional formula and you get the most useful rule in probability: to find the chance of “this and then that”, walk the path and multiply.

A bag holds red and blue balls. Draw one, then draw another. Every branch of the tree carries a conditional probability — after a red ball leaves the bag, the bag has changed. Click any end of the tree to see its path multiplied out, and run a simulation to check the maths against reality.

The Law of Total Probability

What is the chance the second ball is red? It can happen along two different roads — red then red, or blue then red. Add up every road that ends where you want.

In the tree above, press P(2nd red) to see both roads light up. In the student chart of Section 04 the same law drew the dashed line: overall pass rate = weighted average of the group pass rates. This sum is exactly the denominator of Bayes' theorem — which is where we go next.

Flip the Question — Bayes from the Tree

A tree answers forward questions easily: if the first ball is red, how likely is a red second? But often we see the end and want the start: the second ball was red — how likely was the first one red?

Bayes' theorem in one sentence: the road you care about ÷ all the roads that lead to what you saw. In symbols, start from the bridge of Section 03 and divide:

Switch on with replacement and press the Bayes button again: the answer becomes exactly P(1st red). When draws are independent, seeing the second ball teaches you nothing about the first — Section 04 again.

Prior, Likelihood, Evidence, Posterior

Each part of Bayes' theorem has a name, and the names tell the story of how we learn: start with a belief, see evidence, update the belief.

The Medical Test — Think in People, Not Percentages

A disease affects 1% of people. A test catches 99% of sick people, and falsely flags 2% of healthy people. Your test is positive. Are you 99% likely to be sick?

Most people say yes. Let's test 1,000 people and simply count. Then press Only positive tests — that is the crowd you are actually in.

About 10 sick people test positive — but so do about 20 healthy people, because there are 99 times more healthy people to make mistakes on. A positive test means roughly 1 in 3, not 99%. Now raise the prevalence slider to 20% (say, a patient who already has symptoms): the same test suddenly becomes very convincing. The prior changes everything.

Learning From Evidence — Update, Then Update Again

Bayes' theorem is not a one-off calculation; it is a learning loop. Today's posterior becomes tomorrow's prior. Take a second test and update again.

Two positive tests in a row take the belief from 1% to about 96%: the second test starts from a much higher prior (33%). One negative result pulls it sharply back down. This loop — believe, observe, update — is how spam filters, medical diagnosis, self-driving cars and many machine-learning models reason under uncertainty.

Practice — Your Turn

Answer as a decimal (0.62) or a percentage (62%). Draw a tree or picture a crowd of 1,000 first — it makes every problem easy.

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