Set Notation & Venn Diagrams

Set Notation

What is a set?

A set is a collection of elements
- Elements could be anything - numbers, letters, coordinates
You could describe a set by writing its elements inside curly brackets { }
- e.g. {1, 2, 3, 6} is the set of the factors of 6
In probability a subset of possible outcomes can be useful
- e.g. If a whole number between 1 and 50 is chosen at random
  - we might be interested in the outcome being a multiple of 10
  - the set {10, 20, 30, 40, 50}
  - which is a subset of the set {1, 2, 3, ..., 48, 49, 50}

What do I need to know about set notation?

$E$ is the universal set (the set of everything)
- e.g. if talking about factors of 24 then $E$ = {1, 2, 3, 4, 6, 8, 12, 24}
- You may see alternative notations used for $E$
  - U is a common alternative
  - S or the Greek letter ξ (xi) may also be seen
Sets, particularly in probability, are often referred to by capital letters
- e.g. The set A is the set of even factors of 24
- A = {2, 4, 6, 8, 12, 24}
a ∈ B means a is an element of B ( a is in the set B)
A ∩ B means the intersection of A and B (the overlap of A and B)
- the set of elements that are in both sets A and B
- in probability, intersection (∩) generally means AND
A ∪ B means the union of A and B (everything in A or B or both)
- the set of elements that are in either set A, or set B, or both
- in probability, union (∪) generally means OR
A' (called "A prime" but frequently called "A dash") means "not A" (everything outside A)
- the set of elements that are not in set A
You may occasionally see the symbol ∅ - this is the empty set (the set with no elements)

Venn Diagrams

What is a Venn diagram?

Venn diagrams allow us to show two (or more) characteristics of a situation where there is overlap between the characteristics
For example, students in a sixth form can study biology or chemistry but there may be students who study both

What might I be asked to do with a Venn diagram?

You can be asked to
- draw a Venn diagram and/or
- interpret a Venn diagram
Strictly speaking the rectangle (box) is always essential on a Venn diagram
- it represents everything that can happen in the situation
- you may see the letter $E$ or $ξ$ written inside or just outside the box
  - this means “the set of all possible outcomes” - i.e. “everything”!
  - sometimes the letters U or S are used instead
The words AND and OR become very important in both drawing and interpreting Venn diagrams
You will need to be familiar with the symbols ∩ and ∪
- ∩ is intersection
- ∪ is union
- these mean AND and OR (respectively)

How do I draw a Venn diagram?

Start with a “box” and overlapping “bubbles”
- the number of bubbles needed will depend on how many characteristics you are dealing with
- it will usually be 2 or 3
Work through each sentence/piece of information given in a question to begin completing sections of your Venn diagram
- pieces of information may have to be combined before you can enter a value into the diagram
- not all values will be given directly
  - some may need working out
  - you will be expected to do this to complete your Venn diagram
Remember to consider AND and OR

How do I interpret a Venn diagram?

Use the information in the question to identify the parts of the Venn diagram needed to answer it
Shading the relevant parts of a Venn diagram can be helpful
Be careful with probability notation such as
- A' (not A)
- the symbols ∩ and ∪

venn-diagram-2

How do I use Venn diagrams with conditional probability?

Probabilities that only involve a subset of the things in a Venn diagram are called conditional probabilities
- For example with students studying German or Spanish or both, you might want to know the probability that a student who studies Spanish also studies German
- This would require considering only the 'Spanish' part of the Venn diagram when finding the probability
Conditional probability questions are often (but not always!) introduced by the expression 'given that...'
- For example 'Find the probability that a randomly chosen student studies German, given that the student also studies Spanish'
- The answer would be the number in the 'Spanish AND German' part of the diagram divided by the sum of all the numbers in the 'Spanish' part of the diagram
Conditional probabilities are sometimes written using the 'straight bar' notation $P (A | B)$
- That is read as 'the probability of A given B'
- For example $P (German | Spanish)$ would be the probability that a student studies German, given that the student also studies Spanish

Exam Tip

You may have to use your Venn diagram more than once in a question
- so shading the original diagram can become confusing if you're trying to use it more than once
- draw a 'mini'-Venn diagram (a small quick sketch just showing the box and bubbles but no values) and shade that

Worked example

In a class of 30 students, 15 students study Spanish, and 3 of the Spanish students also study German.
7 students study neither Spanish nor German.

Draw a Venn diagram to show this information.

We start with the 3 in the intersection ("overlap"); we can then deduce the "Spanish only" section is 12.
7 needs to be outside both bubbles but within the box.
With a total of 30 we can work out how many students study "German only" and complete the diagram.

Use your Venn diagram to find the probability that a student, selected at random from the class, studies Spanish but not German.

Highlight the part "Spanish only".

Pick out the numbers you need carefully.

Students studying "Spanish only" = 12
Total number of students = 30

P(Spanish only) $= \frac{12}{30} (= \frac{2}{5})$

Worked example

Given the Venn diagram below, find the following probabilities:

P( A)

Draw a 'mini'-Venn diagram - a quick sketch without the details.
See these questions as "ways to win" - so in this part you win if in "bubble A".
B and C do not come into it at all.

Total in A = 3 + 5 + 1 + 8 =17
Total = 3 + 5 + 1 + 8 + 2 + 4 + 8 + 9 = 40

$P (A) = \frac{17}{40}$

P( A ∩ B ∩ C )

∩ - intersection - AND - "win" if "in A" AND "in B" AND "in C".

Total in A AND B AND C = 8
Total = 40

$P (A \cap B \cap C) = \frac{8}{40} (= \frac{1}{5})$

P( B' ∩ C )

∩ - intersection - AND - "win" if "not in B" AND "in C".

Total in "not B" AND C = 5
Total = 40

$P (B' \cap C) = \frac{5}{40} (= \frac{1}{8})$

P( A ∪ B )

∪ - UNION - OR - "win" if "in A" OR "in B".

Total in A OR B = 3 + 1 + 8 + 5 + 2 + 8 = 27
Total = 40

$P (A \cup B) = \frac{27}{40}$

P( A ∪ B ∪ C )

∪ - union - OR - "win" if "in A" OR "in B" OR "in C".

In this case, it will be easier to subtract from the whole total.

Total in A OR B OR C = 40 - 9 = 31
Total = 40

$P (A \cup B \cup C) = \frac{31}{40}$

P( A' ∪ B' )

∪ - union - OR - "win" if "not in A" OR "not in B".
This one is particularly difficult to see without a diagram!

Total in A' OR B ' = 40 - 8 - 5 = 27
Total = 40

$P (A' \cup B') = \frac{27}{40}$

Worked example

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