Set Notation and Venn Diagrams

AS · S1 · 3 min

Probability is written in the language of sets. An event is a set of outcomes, and the words "and", "or", "not" become intersection, union and complement. Venn diagrams make the relationships visible and turn many probability questions into simple arithmetic on regions.

Notation

Key result

Symbol	Read as	Meaning
$A \cap B$	$A$ and $B$	outcomes in both
$A \cup B$	$A$ or $B$	outcomes in at least one
$A'$	not $A$	outcomes not in $A$ (the complement)
$P(A \mid B)$	$A$ given $B$	probability of $A$ knowing $B$ has occurred

Key results:

P(A') = 1 - P(A), \qquad P(A \cup B) = P(A) + P(B) - P(A \cap B)

Events are mutually exclusive if $P(A \cap B) = 0$ ; then $P(A \cup B) = P(A) + P(B)$ .

Venn diagrams

Draw a rectangle for all outcomes and overlapping circles for events. Fill in the intersection first, then the rest of each circle, then what is outside both.

Filling a Venn diagram

In a class of 30, 18 study French, 14 study Spanish and 6 study both. Draw a Venn diagram and find the number who study neither.

Solution

Intersection: $6$ . French only: $18 - 6 = 12$ . Spanish only: $14 - 6 = 8$ . Total in at least one: $26$ . Neither: $30 - 26 = 4$ .

Using the addition rule

$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cup B) = 0.7$ . Find $P(A \cap B)$ , $P(A' \cap B)$ and $P(A' \cap B')$ .

Solution

$P(A \cap B) = 0.5 + 0.4 - 0.7 = 0.2$ . $P(A' \cap B) = P(B) - P(A \cap B) = 0.2$ . $P(A' \cap B') = 1 - P(A \cup B) = 0.3$ .

Three events

Of 100 students, 40 play football, 35 play hockey, 30 play tennis; 12 play football and hockey, 10 football and tennis, 8 hockey and tennis; 5 play all three. How many play none?

Solution

Start from the centre: all three $5$ . Football and hockey only: $12 - 5 = 7$ ; football and tennis only: $5$ ; hockey and tennis only: $3$ . Football only: $40 - 7 - 5 - 5 = 23$ ; hockey only: $35 - 7 - 3 - 5 = 20$ ; tennis only: $30 - 5 - 3 - 5 = 17$ .

Total playing something: $23 + 20 + 17 + 7 + 5 + 3 + 5 = 80$ . None: $20$ .

Translating words

Words	Set	Region
both $A$ and $B$	$A \cap B$	the overlap
$A$ or $B$ (or both)	$A \cup B$	everything inside either circle
$A$ but not $B$	$A \cap B'$	$A$ minus the overlap
exactly one of $A$ , $B$	$(A \cap B') \cup (A' \cap B)$	the two crescents
neither	$A' \cap B'$	outside both circles

Exactly one

$P(A) = 0.6$ , $P(B) = 0.5$ , $P(A \cap B) = 0.3$ . Find the probability that exactly one of $A$ and $B$ occurs.

Solution

$P(A \cap B') + P(A' \cap B) = (0.6 - 0.3) + (0.5 - 0.3) = 0.5$ .

Watch out

" $A$ or $B$ " in probability includes the case where both happen. Do not subtract the intersection twice: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ already removes the double count once.

Exam tip

When a question gives probabilities of $A$ , $B$ and either their union or intersection, draw the Venn diagram with the four regions labelled. Then $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ and independence checks are read straight from it; see Conditional Probability.

Practice

Question

$P(A) = 0.35$ , $P(B) = 0.45$ , $P(A \cap B) = 0.15$ . Find $P(A \cup B)$ and $P(A' \cap B')$ .
50 people: 28 like tea, 30 like coffee, 4 like neither. How many like both?
$A$ and $B$ are mutually exclusive with $P(A) = 0.3$ and $P(B) = 0.5$ . Find $P(A \cup B)$ and $P(A \mid B)$ .
Describe in words the region $A' \cup B$ .

Answers

$0.65$ ; $0.35$ .
At least one: $46$ ; both $= 28 + 30 - 46 = 12$ .
$0.8$ ; $P(A \mid B) = 0$ (they cannot both occur).
Everything except " $A$ but not $B$ ".

Notation

Venn diagrams

Translating words

Practice

Console