J560 · 11.01–11.02ProbabilityFoundation & HigherGrades 4–7

Venn Diagrams & Set Notation

Q: How do I show A^ B?

Shade the part of B that is outside A — the "B only" region.

In a class of 30, 18 play football, 12 play tennis and 5 play both. How many play neither? A $\textbf{Venn diagram}$ answers questions like this at a glance, by drawing overlapping circles for each group. Set notation — symbols such as $\cup$ , $\cap$ and $A^{\prime}$ — describes the regions precisely, and OCR uses it in questions at both tiers.

45 min Probability

The big picture

The universal set $\xi$ is everything being considered, drawn as a rectangle. Each set is a circle inside it. The overlap holds items in both sets; the space outside the circles holds items in neither. Once every region has its number, any probability is a region count divided by the total.

The golden rule is to fill Venn diagrams from the middle outwards: the intersection first, then the "only" parts, then the outside. Starting with the full set totals double-counts the overlap — the most common Venn diagram error.

Venn Diagrams - CorbettmathsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

Before you start

Try a short calculation from the preceding topic without the notes.
Check the tier and grade band shown in this lesson.

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Use set notation: $\xi$ , $\in$ , $\cup$ , $\cap$ and $A^{\prime}$

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: $A = \{2, 4, 6, 8\}$ and $B = \{4, 8, 12\}$ . What is $A \cap B$ ?

Show answer and reasoning

$\{4, 8\}$ . Elements in both sets.

After watching 2: The symbol $\cup$ means:

Show answer and reasoning

Union. Everything in either set.

What you'll learn

Use set notation: $\xi$ , $\in$ , $\cup$ , $\cap$ and $A^{\prime}$
Complete Venn diagrams from given information
Find probabilities from Venn diagrams
Use Venn diagrams with three sets and with algebraic unknowns (Higher)

Set notation

$\xi$ is the universal set. $x \in A$ means " $x$ is a member of set $A$ ".

$A \cap B$ (intersection) is everything in both $A$ and $B$ . $A \cup B$ (union) is everything in $A$ or $B$ or both.

$A^{\prime}$ (complement) is everything not in $A$ .

Sets can be listed in curly brackets: if $\xi = \{1, 2, \dots, 10\}$ and $A$ is the even numbers, $A = \{2, 4, 6, 8, 10\}$ .

Example 1

\xi = \{1, 2, \dots, 12\}

A =

multiples of 3,

B =

factors of 12. List

A \cap B

and

A \cup B

A = \{3, 6, 9, 12\}

B = \{1, 2, 3, 4, 6, 12\}

2Both:

\{3, 6, 12\}

3Either:

\{1, 2, 3, 4, 6, 9, 12\}

Answer

A \cap B = \{3, 6, 12\}

;

A \cup B = \{1, 2, 3, 4, 6, 9, 12\}

Your turnSame method, new numbers — try it before moving on.

$\xi = \{1, 2, \ldots, 10\}$ . $A$ is the even numbers and $B$ is the prime numbers. List the members of $A \cap B$ .

Your answer

Tip — Remember " $\cap$ looks like an n for iNtersection" and " $\cup$ is a cup that holds everything in either set".

Completing a Venn diagram

Start with the intersection. Then subtract it from each set’s total to get the "only" regions. Finally subtract everything inside the circles from the universal total to get the outside region.

Check that all regions add to the total.

Example 130 students: 18 play football (

F

), 12 play tennis (

T

), 5 play both. How many play neither?

1Intersection: 5.

2Football only:

18 - 5 = 13

. Tennis only:

12 - 5 = 7

3Inside circles:

13 + 5 + 7 = 25

Answer

30 - 25 = 5

play neither.

Your turnSame method, new numbers — try it before moving on.

40 students: 25 have a cat, 18 have a dog, and 9 have both. How many have neither?

Your answer

Why this works

Adding $18 + 12 = 30$ suggests everyone plays a sport, but that counts the 5 who play both twice. The Venn diagram makes the double-count visible.

Probability from Venn diagrams

P(region) $=$ number in region ÷ total in $\xi$ .

Using the football example: P( $F \cap T$ ) $= \tfrac{5}{30}$ , P( $F \cup T$ ) $= \tfrac{25}{30}$ , P( $T^{\prime}$ ) $= \tfrac{18}{30}$ .

Diagrams can also be filled with probabilities instead of counts; then all regions sum to 1.

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

The addition rule.

Unknowns and three sets

If a region is unknown, call it $x$ , express the other regions in terms of $x$ , and use the total to form an equation.

With three sets, fill the central region (all three) first, then the regions in exactly two sets, then the "only" regions, then the outside.

Example 1In a group of 50 people, 28 like tea, 25 like coffee, and 7 like neither. How many like both?

1Let

x

like both: tea only

28 - x

, coffee only

25 - x

(28 - x) + x + (25 - x) + 7 = 50

60 - x = 50

, so

x = 10

Answer10 people like both.

Your turnSame method, new numbers — try it before moving on.

60 people: 35 like film X, 30 like film Y, and 10 like neither. How many like both?

Your answer

Tip — Label every region in terms of $x$ on the diagram before writing the equation. It stops you missing a region.

Think like an examiner

Fill Venn diagrams from the intersection outwards.
Check that all regions sum to the total.
Use set notation correctly in answers when asked.

Sets

A \cap B

Intersection: in both.

A \cup B

Union: in either or both.

A'

Complement: not in A.

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

Addition rule.

Watch out for these

Putting 18 in the football-only region when 5 play both.

Football only is

18 - 5 = 13

Confusing

\cup

with

\cap

\cup

is union (either);

\cap

is intersection (both).

Forgetting the region outside the circles.

Items in neither set belong in the rectangle outside.

Reading

A^{\prime}

as "only

A

A^{\prime}

is everything outside

A

, including items in neither set.

Stretch yourself

For events $A$ and $B$ , P( $A$ ) $= 0.5$ , P( $B$ ) $= 0.4$ and P( $A \cap B$ ) $= 0.15$ . Find P( $A \cup B$ ), P( $A^{\prime} \cap B^{\prime}$ ) and P( $A \cap B^{\prime}$ ).

Hint — Draw a Venn diagram with probabilities, starting from the intersection.

Questions students ask

What does ξ mean?

The universal set — everything being considered in the problem.

A \cup B

the same as

A + B

No. The union counts items in both sets only once.

How do I show

A^{\prime} \cap B

Shade the part of

B

that is outside

A

— the "

B

only" region.

Key takeaways

$\cap$ intersection, $\cup$ union, $A^{\prime}$ complement, $\xi$ universal set.
Fill Venn diagrams from the middle out.
Probability $=$ region ÷ total, or read directly from a probability Venn diagram.
P( $A \cup B$ ) $=$ P( $A$ ) $+$ P( $B$ ) $-$ P( $A \cap B$ ).
Use $x$ for unknown regions and form an equation from the total.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 43 marks

$\xi = \{1, 2, \ldots, 15\}$ , $A$ = multiples of 4, $B$ = odd numbers.

(a)Explain why $A \cap B$ has no members.[1]
(b)Find $n(A \cup B)$ .[2]

Question 2Grade 62 marks

$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.15$ . Work out $P((A \cup B)')$ .[2]

Question 3Grade 73 marks

A Venn diagram has regions: $A$ only $2x$ , both $x$ , $B$ only 12, neither 6. The total is 45.

Work out $P(A)$ .[3]

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

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Generating a question…

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OCR J560 official specification (opens in a new tab)
Check the assessed scope and practical or mathematical requirements.
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An external explanation or interactive activity to use alongside the lesson. It may include extension material.

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

40 people: 22 have a cat, 15 a dog, 6 both. How many have neither?
Using the same data, what is P(cat only)?
P( $A$ ) $= 0.6$ , P( $B$ ) $= 0.3$ , P( $A \cup B$ ) $= 0.75$ . What is P( $A \cap B$ )?
Which region is $A^{\prime} \cap B$ ?

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons

Venn Diagrams & Set Notation

In a class of 30, 18 play football, 12 play tennis and 5 play both. How many play neither? A

\textbf{Venn diagram}

answers questions like this at a glance, by drawing overlapping circles for each group. Set notation — symbols such as

\cup

\cap

and

A^{\prime}

— describes the regions precisely, and OCR uses it in questions at both tiers.

45 min Probability

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Use set notation:

\xi

\in

\cup

\cap

and

A^{\prime}

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: $A = \{2, 4, 6, 8\}$ and $B = \{4, 8, 12\}$ . What is $A \cap B$ ?

Show answer and reasoning

$\{4, 8\}$ . Elements in both sets.

After watching 2: The symbol $\cup$ means:

Show answer and reasoning

Union. Everything in either set.

Practise it the way it is examined

Question 1Grade 43 marks

$\xi = \{1, 2, \ldots, 15\}$ , $A$ = multiples of 4, $B$ = odd numbers.

(a)Explain why $A \cap B$ has no members.[1]
(b)Find $n(A \cup B)$ .[2]

Question 2Grade 62 marks

$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.15$ . Work out $P((A \cup B)')$ .[2]

Question 3Grade 73 marks

A Venn diagram has regions: $A$ only $2x$ , both $x$ , $B$ only 12, neither 6. The total is 45.

Work out $P(A)$ .[3]

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

40 people: 22 have a cat, 15 a dog, 6 both. How many have neither?

Using the same data, what is P(cat only)?

A

)

= 0.6

, P(

B

)

= 0.3

, P(

A \cup B

)

= 0.75

. What is P(

A \cap B

Which region is

A^{\prime} \cap B