Foundations — Complementary events — P(A') = 1 − P(A)

Before we can use $P(A^{\prime })=1-P(A)$, we have to earn every symbol in it. This page assumes you have seen nothing. We build the picture first, then attach the notation to that picture, one piece at a time. See the parent note Complementary Events once you finish here.

1. An "outcome" — the smallest thing that can happen

Roll one die. The possible outcomes are "landed on 1", "landed on 2", …, "landed on 6". You cannot split "landed on 3" into anything smaller — it is atomic.

The picture: think of each outcome as one dot. One roll of a die produces six dots, because there are six ways it can land.

Why the topic needs it: everything else — events, complements, probabilities — is built out of these dots. If you can't see the dots, the rest is fog.

2. The "sample space" $S$ — all the dots at once

For a die, $S=\{1,2,3,4,5,6\}$.

The notation, symbol by symbol:

• The curly braces $\{\dots \}$ mean "the set containing" — a fence drawn around a group of things.
• The commas separate the individual outcomes (dots) inside the fence.
• The letter $S$ is just a name for that whole fenced-off group. We could call it anything; $S$ stands for Sample space.

The picture: draw a rectangle. Every dot lives inside it. Nothing that can happen lives outside it — the rectangle is everything that can happen.

Why the topic needs it: the complement of an event means "everything in $S$ except that event." Without a clearly drawn "everything", the word "except" has no meaning. See 1.3.01-Sample-space-and-events for more.

3. An "event" $A$ — a chosen handful of dots

Example: "rolling an even number" is the event $A=\{2,4,6\}$. You circled three of the six dots.

The picture: inside the $S$ rectangle, draw a loop around some dots. Everything you looped is $A$. The loop can hold one dot, several dots, all the dots, or even zero dots.

Why the topic needs it: the whole rule is about an event $A$ and its opposite. $A$ is the "thing I care about" — the loop.

4. Two symbols for combining loops: $\cap$ and $\cup$

We will need to talk about what two loops share and what they cover together.

Memory hook for the shapes:

• $\cap$ looks like a little bridge / cap — think "cap = common = the overlap".
• $\cup$ looks like a cup that scoops up everything — "cup collects it all".

The picture: two overlapping loops. The lens-shaped middle is $\cap$. The whole blob (both loops together) is $\cup$.

Why the topic needs it: the complement is defined by two facts stated with these symbols — the two loops never overlap ($\cap$ is empty) and together they fill the rectangle ($\cup$ is all of $S$).

5. The empty set $\varnothing$ — a loop holding nothing

The picture: an empty loop, containing no dots at all.

Why the topic needs it: when we say $A$ and its opposite "never happen together", we write $A\cap A^{\prime }=\varnothing$ — their overlap contains nothing. $\varnothing$ is how we say "no overlap" in symbols.

6. The complement $A^{\prime }$ — everything the loop left behind

The notation: the little mark ${}^{\prime }$ (a prime, read "$A$-prime") is a switch that flips "inside the loop" into "outside the loop". The three spellings $A^{\prime }$, $A^c$ ($c$ for complement), and $\bar{A}$ (a bar over the top) all mean exactly the same thing.

The picture: shade everything inside the rectangle except the loop. That shaded region is $A^{\prime }$.

Why the topic needs it: this is the star of the show. The rule $P(A^{\prime })=1-P(A)$ is a statement about the shaded region versus the loop.

7. Probability $P(\cdot )$ — how much of the rectangle a loop takes up

Reading the notation: "$P$" is a machine. You feed it an event (put it in the brackets), it hands back a number. $P(A)$ = "the probability of $A$". $P(A^{\prime })$ = "the probability of not-$A$".

The scale, with pictures:

• $P=0$: an empty loop ($\varnothing$) — takes up no area — impossible.
• $P=1$: a loop that is the whole rectangle ($S$) — takes up all the area — certain.
• $P=\frac{1}{2}$: a loop covering half the area — a coin flip.

Why area? Because area is conserved: if the loop grows, the outside shrinks by exactly the same amount, and the total is always the full rectangle. That conservation is precisely the complement rule in disguise.

8. Putting the symbols together — reading the rule out loud

Now every symbol in the parent rule is earned. Read it as a sentence:

$P(A^{\prime })=1-P(A)$

"The area outside the loop equals the whole rectangle ($1$) minus the area inside the loop."

That is it. The addition rule from 1.3.03-Addition-rule-for-probabilities (non-overlapping loops' areas add) plus the axiom $P(S)=1$ turns this obvious area-fact into a formula.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section before touching the parent note.