Foundations — Mutually exclusive events — addition rule

Before you can trust a single line of the parent note, you must be able to read every squiggle in it without flinching. This page builds each one from nothing, in an order where each idea leans on the one before it.

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1 — An "outcome": the atom of everything

Roll a die once. It lands showing a face. That face is the outcome. You cannot split it further; it is the atom.

Why the topic needs it: every probability we compute is really "how many dots are in the group I care about". If you don't see outcomes as countable dots, the addition rule is just symbols.

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2 — The sample space $S$ and the count $n(\cdot )$

Figure 1. The teal rectangle is the whole sample space $S$ for one die roll; each orange dot is a single outcome (faces 1–6). The plum arrow points at one dot to remind you "one outcome = one dot". The label $n(S)=6$ in the corner is the counter reporting six dots total — this is exactly the denominator you will divide by for every probability.

The symbol $n(\cdot )$ is a machine that counts how many dots are inside whatever you put in the brackets:

• $n(S)$ = how many dots total (for a die, $n(S)=6$).
• $n(A)$ = how many dots are in group $A$.

Why the topic needs it: the parent's very first line is $P(E)=\frac{n(E)}{n(S)}$. That whole formula is nothing but "wanted dots over total dots". $n$ is the counter.

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3 — An "event": a fenced-off group of dots

The word subset literally means "a group taken from inside $S$". Every dot in an event is also a dot in $S$; an event never invents new outcomes.

The curly braces $\{\}$ just mean "the collection containing these listed dots".

Why the topic needs it: the addition rule combines two events $A$ and $B$. Everything is a fence around dots.

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4 — Plus $+$ and minus $-$: combine and take away

Before any formula, pin down the two humblest symbols in the whole note.

Why the topic needs it: every formula ahead ($P(A)+P(B)$, $1-P(A)$, $P(A)+P(B)-P(A\cap B)$) is built from just these two moves. If $-$ is a mystery, the general rule is a mystery.

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5 — Probability $P(E)$: fair counting

The letter $P$ is read "the probability of". It always spits out a number between $0$ and $1$:

• $P=0$ means the event has no dots — it can never happen.
• $P=1$ means the event contains all the dots — it always happens.
• $P=\frac{1}{2}$ means half the dots are yours.

Why the topic needs it: the entire addition rule is derived by first counting dots, then dividing everything by $n(S)$ at the end. $P$ is just the counter $n$ with the total divided out.

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6 — Union $\cup$: the word "OR" as a picture

Figure 2. Two overlapping fences: the orange disc is event $A$, the teal disc is event $B$. Everything shaded — orange part, teal part, and the middle lens marked "both" — is the union $A\cup B$. Notice the lens in the centre: those dots belong to both fences, which is exactly why blindly adding "orange dots + teal dots" would count them twice. This picture is the reason the general rule needs a subtraction.

The symbol $\cup$ looks like a cup that scoops up everything from both fences into one bigger fence. Read $\cup$ out loud as "or".

Why the topic needs it: the addition rule answers exactly one question: what is $P(A\cup B)$? — the chance that at least one of the two events happens.

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7 — Intersection $\cap$ and the empty set $\varnothing$

The symbol $\cap$ is an upside-down cup — a cap — and read $\cap$ as "and". It keeps only the shared dots.

Why the topic needs it: "mutually exclusive" is defined as $A\cap B=\varnothing$. That single line says "the two fences share no dots", so their overlap is empty, so there is nothing to double-count.

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8 — Putting it together: the general addition rule

Now that $+$, $-$, $\cup$ and $\cap$ all mean something, we can state the master formula the whole parent note rests on.

Why the topic needs it: this is the rule the parent "earns" the special case from. The mutually-exclusive shortcut is just this formula with the overlap equal to zero.

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9 — Mutually exclusive: fences that never touch

Figure 3. Here the orange fence $A$ and the teal fence $B$ sit apart with a clear gap between them (plum arrow) — no dot is shared, so $A\cap B=\varnothing$. Compare with Figure 2, which had a "both" lens; that lens has vanished. With no overlap to remove, the subtraction term in the general rule is zero and plain addition is exactly right.

Why the topic needs it: this is the entire topic in one condition. "Can both happen at once?" — if no, the overlap is $\varnothing$ and you just add; if yes, subtract the overlap first.

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10 — Complement $A^{\prime }$: everything outside the fence

Picture the whole sample-space rectangle; $A$ is a fenced patch inside it; $A^{\prime }$ is all the rest of the rectangle.

Why the topic needs it: the parent's "handy corollary" and Worked Example 4 ("$P(\text{not green})=1-0.2$") are just the addition rule applied to $A$ and $A^{\prime }$. You cannot understand the shortcut without $A^{\prime }$.

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11 — Sigma $\sum$: "add up many pieces"

The letter $i$ is a counter that walks from $1$ up to $n$; each step you write down $P(A_i)$ and add it on. The big Greek $\sum$ (capital sigma, "S" for Sum) just means "sum all of these".

Why the topic needs it: when many events are pairwise mutually exclusive, plain addition extends to all of them at once, and $\sum$ is the compact way to say "add every single one".

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Prerequisite map

Everything upstream of General addition rule must be solid before the parent note makes sense. If any box confuses you, re-read its section above.

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Equipment checklist

Cover the right-hand side and test yourself out loud.

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Connections

• Mutually exclusive events — addition rule
• Probability — basic definitions & sample space
• Venn diagrams in probability
• Complementary events
• General addition rule & inclusion–exclusion