Foundations — Independence and mutual exclusivity

This page assumes you have seen none of the notation in the parent note. We build every symbol from the ground up, in the order that each one needs the one before it. If a picture defines a thing better than a sentence, we draw it.

0. The stage: outcomes, the sample space, and events

Before any symbol appears, picture an experiment — rolling a die, flipping a coin, drawing a ball. When you do the experiment, exactly one thing happens. That single thing is an outcome.

The picture: a rectangle. Every dot inside is one possible outcome. For a die, six dots labelled 1–6. Nothing can happen outside the rectangle.

Why the topic needs it: you cannot ask "can both events happen?" until you have drawn the space where events live. The rectangle is the sheet of paper everything else is painted on.

The picture: shade some of the dots. "Roll an even number" is the blob covering the dots 2, 4, 6. The letter $A$ points at that blob.

Why the topic needs it: "mutually exclusive" and "independent" are relationships between two blobs. No blobs, no relationship.

1. Probability $P(A)$ — how much of the rectangle a blob covers

The picture: if the rectangle has area 1 and blob $A$ takes up one-third of it, then $P(A)=\frac{1}{3}$.

For equally-likely outcomes, this is just counting: $P(A)=\frac{\text{number of outcomes in }A}{\text{total number of outcomes}}$

Why the topic needs it: both key definitions — $P(A\cap B)=0$ and $P(A\cap B)=P(A)P(B)$ — are statements about these area-numbers. See 1.3.01-Basic-probability-concepts for where this notation is first introduced.

2. Intersection $\cap$ — where two blobs overlap

The picture: draw blob $A$ and blob $B$ so they cross. The lens-shaped sliver in the middle — inside both — is $A\cap B$.

• If the blobs are pulled apart so they touch nowhere, the overlap is empty and $P(A\cap B)=0$. This is exactly mutual exclusivity.
• If they overlap, $P(A\cap B)$ is the area of that lens.

Why the topic needs it: the entire topic is about the size of the overlap. Mutually exclusive = overlap zero. Independent = overlap equals $P(A)\times P(B)$.

3. Union $\cup$ — everything in either blob

The picture: the total shaded region when you paint both blobs. The formula that always works: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Why the subtraction? If you add the two areas, the overlap lens got painted twice. You subtract one copy to fix the double-count. When the blobs don't overlap ($P(A\cap B)=0$), there is nothing to subtract, so $P(A\cup B)=P(A)+P(B)$ — the parent note's mutually-exclusive rule.

4. Conditional probability $P(A\mid B)$ — zooming into a blob

This is the symbol the parent note leans on most, so we go slow.

The picture: once you know $B$ happened, the outcomes outside blob $B$ are impossible — so you throw away the rest of the rectangle and treat $B$ as your new, smaller world. Then you ask what fraction of that new world is also in $A$: that is the overlap lens divided by the size of $B$.

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$

Why this formula? The numerator is the part of the new world (which is $B$) that also lies in $A$ — the lens. The denominator rescales, because $B$ is now "the whole thing," so its probability must count as the full amount. This tool exists to answer "does learning $B$ change my belief about $A$?" — the exact question that separates independence from dependence. See 1.3.02-Conditional-probability.

5. The multiplication dot $\cdot$ and the product $\prod$

The picture of a product of probabilities: areas multiplying is an area itself. If $A$ covers a fraction $P(A)$ of the width and $B$ covers a fraction $P(B)$ of the height of a unit square, then $P(A)\cdot P(B)$ is the little rectangle in the corner — the overlap you'd get if the two blobs ignored each other completely.

Why the topic needs it: for $n$ independent events, probabilities multiply, and $\prod$ is the compact way to write that chain. This is the engine of the 3.1.01-Naive-Bayes-classifier.

6. Set-membership glue: $\in$, $\ne$, $\propto$

Why the topic needs them: these are the connective tissue in every derivation in the parent note.

How the foundations feed the topic

The two destinations on the left of the topic are mutual exclusivity (overlap = 0) and independence (overlap = product). Both are downstream of the same three ideas: area, overlap, and the shrink-to-a-blob move of conditioning. They flow onward into 5.2.03-Probabilistic-graphical-models and 1.3.03-Bayes-theorem. Head back to the parent Independence and mutual exclusivity once these feel solid, or read it in Hinglish.

Equipment checklist

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