Foundations — Conditional probability

Before you can read a single line like $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, you must earn every mark on the page: the letters, the bar, the cap, the fraction, the whole idea of "probability". This page builds them one at a time, from nothing, in the order they lean on each other. Nothing is used before it is drawn.

This is the foundations page for Conditional probability (read it in Hinglish here).

0. The very first picture — a "sample space"

Everything starts with the set of things that could happen. Roll a die: it could land 1, 2, 3, 4, 5, or 6. That full collection of possibilities is the sample space.

The symbol $\Omega$ is not scary — it is literally just a name for "the whole rectangle".

1. Event — a region inside the box

We rarely care about one outcome. We care about groups: "the roll is even", "the email is spam". A group of outcomes is an event.

We name events with capital letters: $A$, $B$, $D$ (disease), $S$ (spam). A letter is just a label pinned to a region.

2. Probability $P(\cdot )$ — the size of a region

Now we can say what $P$ means. $P$ is a machine: feed it a region, it hands back a number between $0$ and $1$ telling you what fraction of the box that region covers.

3. Intersection $\cap$ — the overlap of two regions

Two events can overlap. "Spam" and "contains the word free" share some emails. That shared region has its own symbol.

The symbol $\cap$ looks like a little cap or bridge joining the two — it "caps" what they have in common.

4. Complement $\neg$ — everything outside a region

The parent note uses $\neg D$ ("not disease") and $\neg A$. This is just the rest of the box.

5. The bar $\mid$ — "given", the world-shrinking symbol

This is the heart of the whole chapter, and its picture is the payoff.

When we learn "$B$ happened", we throw away the whole box except $B$. $B$ becomes our new rectangle. We then re-measure $A$ inside that new box.

6. The zero-probability guard

The definition whispers $P(B)>0$. Here is why, with no hand-waving.

7. Sum $\sum$ — adding up disjoint pieces

The parent's Law of Total Probability uses $\sum$. It just means "add these all up".

8. Multiplication $\cdot$ — chaining two restrictions

Rearranging the bar formula gives $P(A\cap B)=P(A\mid B)\cdot P(B)$. The dot $\cdot$ is ordinary multiplication.

9. Independence — when the bar changes nothing

How the foundations feed the topic

Onward links once these are solid: Bayes-Theorem, Law-of-Total-Probability, Independence, Chain-Rule-of-Probability, and applications in Naive-Bayes-Classifier, Hidden-Markov-Models, Bayesian-Inference, Mutual-Information.

Equipment checklist

Cover the right side and answer out loud before revealing.