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Venn diagram problems and probability

Venn Diagram Scenario Based Paper 1 + 2 9 min read

On this page
  1. In this note
  2. Formula bank
  3. Filling a two-set diagram
  4. Three sets: from the centre out
  5. Unknown regions, smallest and largest
  6. Probability from a Venn diagram
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 1.2 (with 8.3 probability): Venn diagram problems
  • 1.2Fill a Venn diagram from a description (two or three sets), including neither.
  • 1.2Use algebra for an unknown region; find the smallest and largest possible overlap.
  • 8.3Find probabilities from a Venn diagram, including "chosen from those who …".

Before you start: set notation and shading.

Formula bank

OverlapLearn it\(n(A\cap B)=n(A)+n(B)-n(A\cup B)\)the counting rule; n(A ∪ B) = total − neither.Use it when two circle totals and a total.
'Only' regionLearn itcircle total − everything else inside that circleUse it when after the overlap.
Smallest overlapLearn it\(n(A)+n(B)-n(\xi)\) (or 0)Use it when "smallest possible number in both".
Largest overlapLearn itthe smaller of n(A) and n(B)Use it when "largest possible".
ProbabilityLearn it\(\dfrac{\text{number in the region}}{\text{number you choose FROM}}\)"chosen from those who …" changes the bottom.Use it when probability from a Venn diagram.

Filling a two-set diagram

Each number in a region is how many are in that region. "23 play football" is the WHOLE football circle: football only PLUS the overlap. So always start with the overlap and subtract outwards.

General methodHow to complete any two-set Venn diagram

  1. In at least one set = total − neither.
  2. Overlap = n(A) + n(B) − (in at least one).
  3. A only = n(A) − overlap; B only = n(B) − overlap.
  4. Neither goes outside the circles. Check all four add to the total.

Example A class of 40: 23 play football (F), 18 play tennis (T), 6 play neither. Complete the Venn diagram.

Overlap first, then the 'only' regions.

  1. at least one: 40 − 6 = 34
  2. both: 23 + 18 − 34 = 7
  3. F only 16, T only 11

Answer 16, 7, 11 and 6 outside

Football and tennis Venn diagramξFT167116
Check: 16 + 7 + 11 + 6 = 40.

Try it yourself

1
42 people: 25 like A, 18 like B, 7 like neither. How many like both?
Show answer

Answer 8

2
A circle total is 20 and the overlap is 6. How many in the 'only' region?
Show answer

Answer 14

Three sets: from the centre out

General methodHow to complete any three-set Venn diagram

  1. Put the number in all three in the centre.
  2. Each pair total (e.g. P∩C = 8): subtract the centre → the 'those two only' region.
  3. Each circle total: subtract its three inner regions → the 'only' region.
  4. Outside = total − everything inside.

Example 50 students: \(n(P)=20\), \(n(C)=22\), \(n(B)=18\), \(n(P\cap C)=8\), \(n(C\cap B)=6\), \(n(P\cap B)=5\), 3 in all three. Complete it.

Centre 3, then pairs, then 'only', then outside.

  1. pairs only: 5, 3, 2
  2. P only 10, C only 11, B only 10
  3. outside: 50 − 44 = 6

Answer see the diagram

Three-set Venn diagramξPCB35321011106
Inside: 44; outside: 6.

Unknown regions, smallest and largest

General methodHow to use x for an unknown region

  1. Call the unknown region x (usually the overlap).
  2. Write every other region in terms of x.
  3. All regions add up to the total: solve.

Example 30 students: 14 do art, 19 do music, 3 do neither. How many do both?

Call the overlap x; regions add to 30.

  1. \((14-x)+x+(19-x)+3=30\)
  2. \(36-x=30\)

Answer \(x=6\)

Smallest and largest overlap. In 50 people, 34 drink tea and 28 drink coffee. Largest overlap: put the smaller circle inside the bigger: 28. Smallest: \(34+28-50=12\).

Try it yourself

1
30 in a group, 20 in A, 15 in B. Smallest possible overlap?
Show answer

Answer 5

Probability from a Venn diagram

General methodHow to find any probability from a Venn diagram

  1. Decide who you are choosing FROM: everyone (the total) or a group ("from those who play football": that circle).
  2. Count the people in the region you want, inside that group.
  3. Probability = that count ÷ the group size.
From the football/tennis diagramProbability
P(football only)\(\tfrac{16}{40}=\tfrac25\)
P(tennis), chosen from the football players\(\tfrac7{23}\)
P(\(F\cup T\))\(\tfrac{34}{40}\)

Try it yourself

1
9 of the 30 cat owners also own a dog. A cat owner is chosen. P(dog)?
Show answer

Answer \(\tfrac3{10}\)

Ways they test it

Type 1 Complete the diagram

"60 people: 35 have a cat, 28 a dog, 4 neither. Complete the Venn diagram." (2 marks)

Spot it circle totals and "neither".

  1. Overlap first.
  2. Only regions, outside.

For example 35, 28, 4 neither, 60

  1. overlap 35 + 28 + 4 − 60 = 7

Answer 28, 7, 21, 4

Type 2 Regions in terms of y

"Regions 2y + 1, y, y + 4, 12 − 2y; 25 students. Find y."

Spot it letters in the regions.

  1. Add all regions = total.
  2. Solve.

For example total 25

  1. \(2y+17=25\)

Answer \(y=4\)

Type 3 Probability

"A cat owner is chosen. Find the probability they also have a dog." (1 mark)

Spot it "chosen from those who".

  1. Group = the cat circle.
  2. overlap ÷ circle total.

Type 4 Smallest / largest

"42 people: 25 speak Urdu, 20 Arabic. Smallest and largest number who speak both?"

Spot it "smallest possible", "largest possible".

  1. Smallest: n(A) + n(B) − total.
  2. Largest: the smaller set.

Type 5 Three sets

"Complete the Venn diagram" with three circles (3-4 marks)

Spot it three sets and 'all three'.

  1. From the centre out.

Mistakes to avoid

  • ✗ Writing 35 in 'C only'✓ 35 − overlapCircle totals include the overlap.
  • ✗ Forgetting the 'neither' number✓ it goes outside the circlesIt is inside ξ.
  • ✗ P(dog | cat) over 60✓ over 35 (the cat owners)Choose from the right group.
  • ✗ Starting a 3-set diagram from the outside✓ start in the centreEvery total includes the inner regions.

Summary

  • Circle totals include the overlap: fill from the middle outwards.
  • \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).
  • Unknown region: x; all regions add to the total.
  • Probability: who are you choosing from?

If you see …

  • "neither"outside the circles.
  • "chosen from those who"that group is the denominator.

Practice

Practice: easy, medium, hard

Easy

1
40 students: 23 play football, 18 tennis, 6 neither. How many play both?
Show working
  1. \(23+18-34\)

Answer 7

2
Same class: P(football only)?
Show working
  1. \(\tfrac{16}{40}\)

Answer \(\tfrac25\)

3
35 students: 20 chess, 12 cards, 9 neither. How many play both?
Show working
  1. \(20+12-26\)

Answer 6

Medium

1
60 people: 35 cat, 28 dog, 4 neither. A person is chosen. P(dog but not cat)?
Guide me

What is it asking? a probability

What do we know? overlap 7, dog only 21

Plan 21 ÷ 60

Show working
  1. overlap 7
  2. dog only 21
  3. \(\tfrac{21}{60}\)

Answer \(\tfrac7{20}\)

2
Same: a cat owner is chosen. P(also has a dog)?
Guide me

What is it asking? a probability

What do we know? choose from 35 cat owners

Plan 7 ÷ 35

Show working
  1. \(\tfrac7{35}\)

Answer \(\tfrac15\)

3
42 people: 25 speak Urdu, 20 Arabic. Smallest possible number who speak both?
Guide me

What is it asking? the smallest overlap

What do we know? 25 + 20 = 45 > 42

Plan 45 − 42

Show working
  1. \(25+20-42\)

Answer 3

Hard

1
Regions: A only 2y + 1, both y, B only y + 4, outside 12 − 2y; 25 in total. Find n(A).
Guide me

What is it asking? n(A)

What do we know? regions in y

Plan add them = 25, then A only + both

Show working
  1. \(2y+17=25\Rightarrow y=4\)
  2. \(9+4=13\)

Answer 13

PartAnswer / workingMarksGuidance
\(2y+17=25\)M1
y = 4A1
13A1
Total: 3 marks
2
60 students: n(A) = 25, n(B) = 22, n(C) = 27, n(A∩B) = 10, n(A∩C) = 7, n(B∩C) = 9, 4 in all three. How many in none?
Guide me

What is it asking? none

What do we know? centre 4

Plan pairs only, then only regions, then subtract from 60

Show working
  1. pairs only: 6, 3, 5
  2. only: 12, 7, 15
  3. \(60-52=8\)

Answer 8

PartAnswer / workingMarksGuidance
pairs only 6, 3, 5B1
only 12, 7, 15B1
8B1
Total: 3 marks
3
30 students: 14 art, 19 music, 3 neither. How many do music only?
Guide me

What is it asking? music only

What do we know? overlap unknown

Plan x = 6, then 19 − x

Show working
  1. \(36-x=30\Rightarrow x=6\)
  2. \(19-6\)

Answer 13

PartAnswer / workingMarksGuidance
equation in xM1
x = 6A1
13A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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