Listing Outcomes Systematically

SPEC P6, P7 FOUNDATION + HIGHER

Sample spaces, two-way tables, grids, and Venn diagrams — the tools for counting outcomes and computing probabilities

Why List Outcomes Systematically?

To find a theoretical probability you need (favourable outcomes) ÷ (total outcomes) — which means you have to count both. For combined experiments, the count is easy to get wrong unless you're systematic.

Possibility Space (Spec P7)

A possibility space is the complete list of all equally likely outcomes of an experiment. AQA requires you to build possibility spaces for both single experiments and combined experiments.

Single experiment

Roll one fair die. Possibility space:

{1, 2, 3, 4, 5, 6}

6 equally likely outcomes.

Combined experiment

Roll two fair dice. Possibility space has pairs:

{(1,1), (1,2), ... (6,6)}

6 × 6 = 36 equally likely outcomes.

Four enumeration tools (P6)

1. Sample space grids for combined experiments 2. Two-way tables for two-category data 3. Venn diagrams for overlapping events 4. Tree diagrams for sequential events (→ covered in Lesson 5.03). Throughout, we use set notation ( , , , n()) to describe regions precisely.

Sample Space Grid: Two Dice

For two dice, build a 6 × 6 grid. Put one die on each axis and fill each cell with the outcome you care about — the sum, difference, or product of the pair.

Sum of two fair dice (36 outcomes)

+123456

1234567

2345678

3456789

45678910

567891011

6789101112

Reading probabilities from the grid

Every cell shows one equally likely outcome. P(event) = cells that satisfy the event ÷ 36.

P(sum = 7) = 6/36 = 1/6 — the main diagonal of 7s.

P(sum = 2) = 1/36 — only the (1,1) cell.

P(sum = 12) = 1/36 — only the (6,6) cell.

Same grid, different operation

Worked Example — P(sum ≥ 10)

Ex

Two fair dice are rolled. Find P(sum ≥ 10).

M — Method: Highlight every cell in the grid where the sum is 10, 11, or 12. Count the highlighted cells and divide by 36.

+123456

1234567

2345678

3456789

45678910

567891011

6789101112

C — Calculate: Highlighted cells: 3 tens, 2 elevens, 1 twelve = 6 cells.

P(sum ≥ 10) = 636 = 16

E — Evaluate: P(sum ≥ 10) = 1/6. Same as rolling a 6 on one die — makes sense given how few pairs sum to 10+.

Two-Way Tables

A two-way table counts how many items fall into each combination of two categories. Every row and every column has a total, and all four inner cells sum to the grand total.

50 people asked about tea (T) and coffee (C)

Likes TeaDoes not like TeaTotal

Likes Coffee151025

Does not like Coffee20525

Total351550

Row and column totals must agree

Row totals: 25 + 25 = 50. Column totals: 35 + 15 = 50. Inner cells: 15 + 10 + 20 + 5 = 50. All agree — the table is internally consistent.

Worked Example — Reading a Two-Way Table

Ex

Using the tea/coffee table, a person is chosen at random. Find:

(a) P(likes tea) — use the tea column total: 35 out of 50.

P(likes tea) = 3550 = 710

(b) P(likes both) — use the single inner cell for tea AND coffee: 15.

P(both) = 1550 = 310

(c) P(likes neither) — use the cell for no-tea and no-coffee: 5.

P(neither) = 550 = 110

Reading the right cell

Translate the phrase first, then point to the cell. "Tea only" → likes tea AND does not like coffee. "At least one" → every cell except neither. "Both" → the single tea-and-coffee cell.

Venn Diagrams

A Venn diagram shows how two (or more) events overlap. Overlapping circles sit inside a rectangle that represents every item being considered — the universal set.

Four regions in every 2-circle Venn

A only — inside A, outside B.

Both — inside both circles (the overlap).

B only — inside B, outside A.

Neither — outside both, but still inside the rectangle U.

Link to 5.01 — Mutually exclusive events

If A and B are mutually exclusive, they can't both happen — so the "Both" region has 0 elements. On a well-drawn Venn, the circles can even be drawn as separate (disjoint) circles with no overlap at all.

Set Notation

Set notation lets you describe any Venn region in a single short symbol. These symbols appear in AQA exam questions and mark schemes.

A B

Union — items in A or B or both. The entire shaded part of any diagram that includes the overlap.

A B

Intersection — items in both A and B. Just the overlap region.

A

Complement of A — items not in A. Everything outside the A circle, including the "neither" region.

n(A)

Number of elements in A — a count, not a probability.

or U

Universal set — everything under consideration (e.g. all 30 students in the class).

Probability from a Venn

P(event) = n(region)n(U)

Count the elements in the region you want, then divide by the total in the universal set.

The Four Regions in Set Notation

Combine the symbols to name each region precisely. This is exactly the wording AQA uses in mark schemes.

Region in wordsSet notationWhat it means

A onlyA BIn A but not in B

B onlyA BIn B but not in A

BothA BIn A and in B

Neither(A B)Not in A or B — outside both circles

At least one of A or BA BIn A, in B, or in both

Read the phrase carefully

"Likes A" means A B and A only — any person inside the A circle, including the overlap. "Likes A only" means just A B — excluding the overlap.

Filling a Venn Diagram

AQA questions often give you the totals for A and B and the overlap. You must work out each individual region before you can calculate probabilities.

The golden rule: place the intersection first

Place n(A B) in the overlap first.

Only A = n(A) − n(A B). Put it in the A-only region.

Only B = n(B) − n(A B). Put it in the B-only region.

Neither = n(U) − (sum of the other three regions). Put it outside both circles.

Check: all four regions should sum to n(U).

The most common Venn error

Students write the total n(A) = 18 in the A-only region, not realising that the 18 already includes the overlap. The A-only region is n(A) minus the overlap.

MCE Worked Example — Class of 30

Ex

In a class of 30, 18 play football (F), 14 play tennis (T), and 8 play both. A student is chosen at random. Find P(plays football only) and P(plays neither).

M — Method: Build the Venn diagram. Place n(F T) = 8 first, then subtract to find the individual regions.

C — Calculate the regions:

Only F = 18 − 8 = 10

Only T = 14 − 8 = 6

Both = 8

Neither = 30 − (10 + 8 + 6) = 6

Check: 10 + 8 + 6 + 6 = 30. ✓

E — Evaluate:

P(football only) = 1030 = 13, P(neither) = 630 = 15

Probability from a Venn Diagram

Once the Venn is filled, every probability is count in region ÷ total. Always write the probability using set notation — AQA mark schemes use this form.

Event in wordsProbability notationWorkingAnswer

Plays footballP(F)(10 + 8)/30 = 18/303/5

Plays football onlyP(F T )10/301/3

Plays bothP(F T)8/304/15

Plays at least oneP(F T)(10 + 8 + 6)/30 = 24/304/5

Plays neitherP((F T) )6/301/5

Sense-check with the complement rule (5.01)

P(F T) + P((F T) ) = 4/5 + 1/5 = 1. ✓ "At least one" and "neither" are complements — they must sum to 1.

Avoiding Double-Counting: n(A B)

If you want to count people in A OR B WITHOUT drawing a Venn, you might think "just add n(A) + n(B)". That double-counts anyone in both. The formula corrects this.

n(A B) = n(A) + n(B) − n(A B)

Why the subtraction

When you add n(A) + n(B), the overlap is counted twice — once in each total. Subtract it once so it's counted once.

Ex

Applying to the class of 30

n(F) = 18, n(T) = 14, n(F T) = 8.

n(F T) = 18 + 14 − 8 = 24

Check against the Venn: Only F + Both + Only T = 10 + 8 + 6 = 24. ✓

Shading Venn Regions

AQA often asks you to shade a region matching a given set notation. The trick is to read the symbol precisely before you shade anything.

Entire A circle

Only the overlap

Everything outside A

Exam strategy

Put a finger on every region of the empty Venn, then ask "does this region satisfy the symbol?" Shade only if yes. This stops you from shading B only when you should have shaded B .

Set notationEquivalent region in wordsWhat to shade

A BA onlyPart of A not in B

(A B)NeitherEverything outside both circles

A BNeither — equivalent to (A B)Same as above (De Morgan)

MCE Practice Question

In a group of 40 students, 22 study History (H), 18 study Geography (G), and 10 study both subjects.

(a) Complete a Venn diagram showing the number of students in each region. [2 marks]

(b) A student is chosen at random. Find P(studies only History). [1 mark]

(c) Find P(studies neither). [1 mark]

Total: [4 marks]

M

Place n(H G) = 10 in the overlap first. Then subtract to get H only and G only. Then subtract the three regions from 40 to get "neither".

C

Only H = 22 − 10 = 12. Only G = 18 − 10 = 8. Both = 10. Neither = 40 − (12 + 8 + 10) = 10.

Check: 12 + 10 + 8 + 10 = 40. ✓

E

(a) Venn filled as above.

(b) P(only H) = 1240 = 3/10.

(c) P(neither) = 1040 = 1/4.

Examiner Alert — "A" vs "A only"

Subtle but costly

These two phrases describe different regions:

"Plays football" means anyone inside the F circle — n(F) = Only F + Both.

"Plays football only" means only the F-only region — excludes the overlap.

PhraseSet notationCount (class of 30)

Plays footballF18 (includes the 8 who do both)

Plays football onlyF T10 (excludes the 8 who do both)

Plays at least one of F or TF T24

Plays exactly one of F or TOnly F + Only T10 + 6 = 16

Highlight the keyword

Before you count, underline the crucial word: "only", "exactly", "at least". Each changes which regions you include.

Examiner Alert — Double-Counting

Error 1: double-counting the overlap

Adding n(A) + n(B) for "everyone in A or B" counts the overlap twice.

Class of 30: wrong answer: 18 + 14 = 32 (impossible — more than 30 students). Correct: 18 + 14 − 8 = 24.

Error 2: writing totals in the wrong region

Students write n(F) = 18 in the F-only region. That's wrong — 18 already includes the 8 who play both. F only = 18 − 8 = 10.

Error 3: forgetting the "neither" region

Students fill the three circle regions and stop. Remember: anyone outside both circles still counts towards n(U). Always check that all four regions sum to n(U).

The bulletproof method

(1) Place the overlap first. (2) Subtract to get A only and B only. (3) Subtract the total of the three filled regions from n(U) to get neither. (4) Verify all four regions sum to n(U).

✓ Checkpoint & Common Errors

Core Skills

Build a sample space grid for two combined experiments (e.g. two dice) and count favourable cells.

Read a two-way table for the intersection, union, and complement counts.

Fill a Venn diagram using the subtract-the-intersection method.

Use set notation : (union), (intersection), (complement), n( ) (count).

Compute probabilities as n(region) ÷ n(U) from any of these tools.

Apply the inclusion-exclusion formula : n(A B) = n(A) + n(B) − n(A B).

Shade any region described in set notation.

Students often writeCorrect version

n(A B) = n(A) + n(B)= n(A) + n(B) − n(A B)

Put 18 in F-only regionF only = 18 − overlap

"A" = "A only""A only" excludes the overlap

Forget the "neither" regionCheck all four regions sum to n(U)

Shade P Q for PP is everything outside P

Next lesson

Lesson 5.03 extends listing outcomes to tree diagrams for independent events — multiplying along paths and adding between paths to find combined probabilities.