Set notation for events

Writing events with set notation — intersection ∩, union ∪ and complement ′ — matching each to a region of a Venn diagram, and the probability rules that go with them (P(A′) = 1 − P(A), P(A ∩ B′) = P(A) − P(A ∩ B)).

Statistics · Probability

Set notation for events

Why it works

An event is just a set of outcomes, so the language of sets gives a precise, compact way to describe combinations of events. Three symbols do almost all the work:

Intersection $A \cap B$ — outcomes in both $A$ and $B$ (" $A$ and $B$ ").
Union $A \cup B$ — outcomes in $A$ or $B$ or both (" $A$ or $B$ ").
Complement $A'$ — outcomes not in $A$ ("not $A$ ").

These build up into any region you like.

A \cap B'

is "in

A

but not

B

(A \cup B)'

is "in neither

A

nor

B

A' \cap B'

is the same region (this is one of De Morgan's laws,

(A \cup B)' = A' \cap B'

). The reliable way to read a complicated expression is to shade it on a Venn diagram one piece at a time.Each symbol comes with a probability rule:

Complement: $P(A') = 1 - P(A)$ — the outcomes split into $A$ and not- $A$ .
"Only $A$ ": $P(A \cap B') = P(A) - P(A \cap B)$ — take all of $A$ and remove the

overlap with

B

Union (the addition rule): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ — the overlap

is counted twice when you add, so subtract it once.

Filling in a Venn diagram is usually the cleanest method: work out the intersection first, then subtract to get the "only

A

" and "only

B

" regions, and finally the "outside" region so all four add to

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