Foundations — Independent events — multiplication rule

This page is the toolbox. The parent note (Independent Events) throws around symbols like $P(A)$, $A\cap B$, $P(A\mid B)$, and $\prod$. Before you touch that note, you should be able to see every one of them. We build each from nothing, in the order they depend on each other.

1. An outcome, an event, and the sample space

Everything starts with a random experiment: flip a coin, roll a die, draw a marble.

Picture the sample space as a box holding every possible outcome. An event is a region drawn inside that box — a fence around the outcomes that count as "yes".

Why the topic needs this: you cannot say "$A$ and $B$ both happen" until $A$ and $B$ are regions in the same box. The whole multiplication rule is about how two such regions overlap.

2. The symbol $P()$ — probability as a fraction of the box

Why the topic needs this: the multiplication rule multiplies these fractions. If you don't feel $P(A)$ as "how much of the box", the product $P(A)P(B)$ is just meaningless symbols.

3. Two events at once: $\cap$ (AND) and $\cup$ (OR)

Now put two fenced regions, $A$ and $B$, in the same box. They can overlap.

Why the topic needs this: the entire rule is about $P(A\cap B)$ — the probability of the overlap. "$P(\text{both heads})$" is $P(H_1\cap H_2)$. Get the overlap picture wrong and every example collapses.

4. Conditional probability $P(A\mid B)$ — shrinking the box

This is the single most important symbol to internalise, because the whole rule is derived from it.

Look at the figure: the accent-red slice is $A\cap B$. Before conditioning we compared it to the whole box; after conditioning we compare it only to $B$ (the shaded region). Same slice, bigger fraction, because the denominator shrank.

Why the topic needs this: independence is defined by this symbol. Two events are independent exactly when learning $B$ doesn't shrink-and-reshape things — i.e. $P(A\mid B)=P(A)$. The parent note rearranges the boxed formula above to build the multiplication rule, so you must own it.

5. The independence idea in one picture

Why the topic needs this: this is the destination. Everything else on this page exists so that this line reads as obvious, not magical.

6. The complement $A^c$ and $1-P(A)$

Why the topic needs this: the "at least one" trick. "At least one thing happens" is hard to count directly, but its complement — "none happen" — is one clean product. So the topic writes $P(\text{at least one})=1-P(\text{none}).$

7. The product symbol $\prod$ — repeated multiplication

Why the topic needs this: for many independent events the parent writes $P(A_1\cap \cdots \cap A_n)={\prod }_{i}P(A_i)$ and $P(\text{at least one})=1-{\prod }_i(1-P(A_i))$. Without $\prod$ these would be endless "$\cdots$" chains.

How these foundations feed the topic

Equipment checklist

Cover the right side and answer out loud before opening the parent note.

Connections

• Conditional Probability — Section 4 is the seed the parent's derivation grows from.
• Complement Rule — Section 6, powering the "at least one" shortcut.
• Mutually Exclusive Events — the overlap $A\cap B$ being empty; contrast with independence.
• General Multiplication Rule — what Section 4 becomes before independence is imposed.
• Bayes' Theorem — also built on the conditional-probability box-shrinking idea.
• Binomial Distribution — stacks many independent trials using the $\prod$ of Section 7.