Venn diagrams

Using a Venn diagram to organise events: the intersection A∩B ("and"), the union A∪B ("or"), and the complement A′ ("not"); filling in the regions from the numbers in a problem (starting with the overlap); reading probabilities off the diagram; and the addition formula P(A∪B) = P(A) + P(B) − P(A∩B).

Statistics · Probability

Venn diagrams

Why it works

A Venn diagram draws each event as a circle inside a rectangle (the universal set

\xi

, all possible outcomes). Where circles overlap, outcomes belong to both events. This turns "and / or / not" into regions you can simply count.

The three set operations:

Intersection $A \cap B$ — outcomes in both $A$ and $B$ (the overlap).
Union $A \cup B$ — outcomes in $A$ or $B$ (or both) — everything inside

either circle.

Complement $A'$ — outcomes not in $A$ — everything outside circle $A$ .

Below,

60

students study French (

F

35

study German (

G

), and

20

study both, out of

100

:Fill the overlap first. The numbers in a Venn diagram are the outcomes in that region only, so they add up to the total. Always start with the intersection, then work outwards:

both $= 20$ ,
French only $= 60 - 20 = 40$ ,
German only $= 35 - 20 = 15$ ,
neither (outside both) $= 100 - (40 + 20 + 15) = 25$ .

Now any probability is just a region total over

100

. For example

P(\text{neither}) = \tfrac{25}{100} = 0.25

, and

P(F \cup G) = \tfrac{40+20+15}{100} = 0.75

.

The addition formula. Adding

P(F)

and

P(G)

counts the overlap twice, so

P(A \cup B) = P(A) + P(B) - P(A \cap B).

Here

P(F) + P(G) - P(F\cap G) = 0.60 + 0.35 - 0.20 = 0.75

— the same

0.75

. Subtract the intersection exactly once to undo the double-count.

Useful regions. "Exactly one of

A

B

" is the two only parts added:

P(A) + P(B) - 2P(A\cap B)

. "Neither" is

P((A\cup B)') = 1 - P(A\cup B)