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Probability

Probability

Sample spaces and Venn diagrams

Sample Spaces Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Sample space diagrams
  4. Venn diagrams
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 8.3: sample spaces and Venn diagrams
  • 8.3List every outcome of two events in a sample space diagram.
  • 8.3Read probabilities from a Venn diagram, including \(P(A\cap B)\) and \(P(A\cup B)\).
  • 8.3Fill in a Venn diagram from the information given.

Before you start: Probability of an event and Venn diagrams in Sets. Next: Tree diagrams.

Formula bank

Sample spaceLearn it\(P=\dfrac{\text{cells you want}}{\text{all cells}}\)two dice: 6 × 6 = 36 cells.Use it when two things happen together.
And (Venn)Learn it\(P(A\cap B)\) = the overlapUse it when 'both'.
Or (Venn)Learn it\(P(A\cup B)\) = everything inside A or B (or both)Use it when 'A or B', 'at least one'.
NotLearn it\(P(A')\) = everything outside AUse it when 'not A'.

Sample space diagrams

When two things happen (two dice, a dice and a coin), list every combination in a grid. Each cell is one equally likely outcome. The grid below shows the totals of two dice: the six 7s are underlined.

+123456
1234567
2345678
3456789
45678910
567891011
6789101112

General methodHow to answer any sample-space question

  1. Draw the grid: one event across the top, the other down the side.
  2. Fill each cell with the result (total, product, pair).
  3. Count the cells you want; divide by the total number of cells.

Example Two fair dice are rolled. Find P(total of 10 or more).

Count the cells with 10, 11 or 12.

  1. 10: 3 cells, 11: 2 cells, 12: 1 cell
  2. \(\tfrac{6}{36}\)

Answer \(\tfrac16\)

Try it yourself

1
Two dice. P(total of 8)? (fraction)
Show answer

Answer 5/36

2
Two dice. P(the product is odd)? (fraction)
Hint
Odd × odd only: 3 × 3 cells.
Show answer

Answer 1/4

Venn diagrams

40 students: football and cricketξFC16699
40 students: 22 play football (F), 15 play cricket (C), 6 play both.

Each number counts the people in that region. For a probability, divide by the TOTAL (40). Both: \(\tfrac{6}{40}\). Football or cricket (or both): \(\tfrac{16+6+9}{40}=\tfrac{31}{40}\). Neither: \(\tfrac{9}{40}\).

General methodHow to fill in a Venn diagram

  1. Start with the overlap (both).
  2. Each set's 'only' region = its total − the overlap.
  3. Outside = grand total − everything inside.
  4. Check: all four regions add to the total (or to 1 for probabilities).

Example P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.15. Find P(A ∪ B) and P(neither).

Overlap first.

  1. A only \(=0.5-0.15=0.35\); B only \(=0.4-0.15=0.25\)
  2. \(P(A\cup B)=0.35+0.15+0.25=0.75\)
  3. neither \(=1-0.75\)

Answer 0.75 and 0.25

Try it yourself

1
From the football/cricket diagram: P(plays cricket only)? (fraction)
Show answer

Answer 9/40

2
P(does not play cricket)? (fraction)
Hint
Everything outside C: 16 + 9.
Show answer

Answer 5/8

Ways they test it

Type 1 Complete the sample space

"Complete the table… Find the probability that the total is 7." (3 marks)

Spot it a partly filled grid.

  1. Fill every cell.
  2. Count, divide by the cells.

Type 2 Complete the Venn diagram

"Complete the Venn diagram." (2 marks)

Spot it totals for each set and 'both'.

  1. Overlap first, then 'only', then outside.

For example 22 F, 15 C, 6 both, 40

  1. 16, 6, 9

Answer outside 9

Type 3 Probability from a Venn diagram

"Find \(P(A\cup B')\)." (1-2 marks)

Spot it set notation.

  1. Shade the region.
  2. Add its numbers; divide by the total.

Mistakes to avoid

  • ✗ Putting the set total (22) in the 'only' region✓ 22 − 6 = 16The overlap is part of the 22.
  • ✗ Dividing by the set total instead of the grand total✓ ÷ everyone (40)Unless the question says 'given that' (see conditional probability).
  • ✗ Listing (3, 4) and (4, 3) as one outcome✓ they are different cells

Summary

  • Sample space: grid of every outcome; cells wanted ÷ all cells.
  • Venn: overlap first; only = total − overlap; outside last.
  • ∩ = both (overlap), ∪ = either, ′ = outside.

If you see …

  • two dice / two spinnerssample space grid.
  • 'both', 'neither'Venn diagram.

Practice

Practice: easy, medium, hard

Easy

1
Two dice. P(total of 7)? (fraction)
Show answer

Answer 1/6

2
Two dice. P(a double)? (fraction)
Show answer

Answer 1/6

3
Football/cricket diagram: P(plays both)? (fraction)
Show answer

Answer 3/20

Medium

1
Two dice. P(the two numbers differ by 2)? (fraction)
Guide me

What is it asking? a probability

What do we know? (1,3),(2,4),(3,5),(4,6) both ways

Plan 8 cells of 36

Show working
  1. \(\tfrac{8}{36}\)

Answer 2/9

2
Football/cricket: P(football or cricket)? (fraction)
Guide me

What is it asking? a probability

What do we know? inside either circle

Plan 16 + 6 + 9

Show working
  1. \(\tfrac{31}{40}\)

Answer 31/40

3
A fair spinner (1, 2, 3) and a coin. P(3 and heads)? (fraction)
Guide me

What is it asking? a probability

What do we know? 3 × 2 = 6 outcomes

Plan one of them

Show working
  1. \(\tfrac16\)

Answer 1/6

Hard

1
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. P(A ∪ B)?
Guide me

What is it asking? a probability

What do we know? A only 0.35, B only 0.25

Plan add the three regions

Show working
  1. 0.75

Answer 0.75

PartAnswer / workingMarksGuidance
0.35 and 0.25M1
0.75A1
Total: 2 marks
2
Same: P(neither)?
Guide me

What is it asking? a probability

What do we know? 1 − P(A ∪ B)

Plan 1 − 0.75

Show working
  1. 0.25

Answer 0.25

PartAnswer / workingMarksGuidance
0.25B1
Total: 1 mark
3
Football/cricket: P(neither)? (fraction)
Guide me

What is it asking? a probability

What do we know? 40 − 31

Plan \(\tfrac{9}{40}\)

Show working
  1. \(\tfrac{9}{40}\)

Answer 9/40

PartAnswer / workingMarksGuidance
9/40B1
Total: 1 mark

Can I do this?

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

Quiz

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