Foundations — Conditional probability — P(A - B) = P(A∩B) - P(A)

This page builds every mark used in the parent topic from nothing, in the order they lean on each other. We start with a single dot and finish, symbol by symbol, at the point where the parent's formula becomes something you can see rather than memorise. Nothing below is used before it is drawn and defined.

0. The starting picture: outcomes as dots

Everything begins with stuff that can happen. Imagine an experiment — rolling a die, drawing a card. Each thing that could be the result is one outcome. Draw every outcome as a single dot (figure s01).

Why start here? Because probability is counting dots. If you don't know what a dot is, you can't count them, and every later symbol is secretly a count.

1. Sample space $S$ — the whole box

• Plain words: every possible result, gathered together.
• Picture: the outer rectangle holding all the dots (figure s01).
• Why the topic needs it: conditional probability is about shrinking this box. You can't shrink a box you never drew.

See Sample space and events for the full treatment. For us, $S$ is just "the universe of dots."

2. Event — a chosen circle of dots

• Plain words: a yes/no question about the result, e.g. "is the die score even?" The event is the set of dots that answer yes.
• Picture: a loop enclosing some dots inside $S$ (figure s02).
• Why the topic needs it: the two events named $A$ and $B$ in the parent are the whole story. Everything is about how two circles of dots overlap.

3. The counting symbol $n(\cdot )$ — how many dots

• Plain words: how many dots the circle catches.
• Picture: put your finger on each dot inside the loop and count.
• Why the topic needs it: the parent's derivation starts from raw counts $n(B)$ and $n(A\cap B)$ before turning them into probabilities.

4. Two loops together: union $\cup$ and overlap $\cap$

When two loops share the picture, there are exactly two new ideas — "either loop" and "both loops." We name both now so neither surprises you later.

• Why the topic needs $\cap$ specifically: conditional probability asks a two-part question — "did $B$ happen, and was it also $A$?" Only the overlap answers both parts. Neither loop alone does, and the union $\cup$ answers the weaker "either" question.

5. Probability $P(\cdot )$ — fraction of the box

Now we turn counting into probability. This is the single most important translation.

• Plain words: out of all the ways things can go, what share falls inside $A$?
• Picture: shade the loop; $P(A)$ is how much of the rectangle got shaded — a number between $0$ and $1$.
• Why divide, not just count? Because raw counts aren't comparable across experiments. Dividing by $n(S)$ rescales everything onto a $0$-to-$1$ ruler, where $0=$ never and $1=$ always.

6. The bar "$\mid$" — the word given

This is the symbol the whole topic is named after. Everything before it has been earned, so now we may write it.

• Plain words: "the chance of $A$, now that we know $B$ came true."
• Picture: we erase every dot outside $B$. The box literally shrinks down to just loop $B$ — that becomes our new universe. Then we ask what share of this smaller box is also $A$ (figure s04).
• Why the topic needs it: this is the entire idea. Without the bar there is no "learning something," and probability never updates.

Because the new box is $B$, its size — the number we must divide by — is the count $n(B)$. Reading straight off figure s04, the fraction of the shrunken box that is also $A$ is: $P(A\mid B)=\frac{\text{overlap dots}}{\text{dots left in the shrunken box}}=\frac{n(A\cap B)}{n(B)}.$

This is exactly why the denominator is $P(B)$ and not $P(A)$: $B$ is the shrunken box we are counting inside.

7. The complement $A^c$ — everything not $A$

The parent uses $A^c$ and $B^c$ (in the false-positive example and the complement law), so we earn them too.

• Plain words: "$A$ did not happen."
• Picture: the whole rectangle except loop $A$ is shaded (figure s05).
• Why the topic needs it: the true complement law $P(A\mid B)+P(A^c\mid B)=1$ says inside the fixed box $B$, either you're in $A$ or you're not — the shares must fill the box.

8. The restriction $P(B)>0$ — the box can't be empty

• Picture: an empty rectangle — nothing to count, nothing to condition on.
• Why the topic needs it: it's the one legal fine-print on the whole formula.

How these feed the topic

The diagram below is a dependency map: read it upward. Each box is one symbol from this page; each arrow means "you need the lower idea before the higher one makes sense." Dots build a box, loops make events, counting makes probability, and the bar shrinks the box — out of which conditional probability drops.

How to read it: if you can trace a path from every bottom box up to "Conditional P of A given B" and explain each arrow out loud, you are ready for the parent topic.

Equipment checklist

Test yourself: cover the right side and answer before revealing. If any fail, re-read that section.

Connections

• Sample space and events — the box and loops built here.
• Tree diagrams — another way to picture shrinking boxes step by step.
• Multiplication rule of probability — what you get by rearranging the conditional formula.
• Independence of events — the special case where shrinking the box changes nothing.
• Bayes' Theorem — safely swapping which event is "given."
• Law of total probability — adding up conditionals across a partition of the box.