Listing outcomes with tables, grids and Venn diagrams
Most probability mistakes are counting mistakes. A systematic listing method removes the guesswork and makes the missing outcome visible.
Part of your national curriculum
- Probability: Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams
Lesson overview
What you'll learn in this lesson
Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams
Key learning points
- • Sample space diagrams
- • Two-way tables
- • Venn diagrams
- • Set notation
This lesson at a glance
- 28 minutes
- 16 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 16
Watch first
Listing outcomes with tables, grids and Venn diagrams
An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.
In a nutshell
Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams
Learning cycle
Part 2 of 16
Learning cycle 1 of 2
Part 1 · Sample space diagrams
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 16
Learn
Sample space diagrams
For two dice, a 6 by 6 grid shows all 36 equally likely outcomes. Reading off the totals gives P(total 7) = 6/36 = 1/6, because six cells run along the diagonal. Listing pairs without a grid nearly always misses cases such as (2, 5) and (5, 2) being different outcomes.
Explore the idea
Part 4 of 16
Learn
Two-way tables
A two-way table cross-classifies data, for example subject choice against year group. Row and column totals must agree with the grand total, and probabilities are read directly as the relevant cell over the relevant total, so choosing the right total is the crucial decision.
Quick check
Part 5 of 16
Quick check
Rolling two dice, P(total 7) is
Quick check
Part 6 of 16
Quick check
A ∩ B means
Reset break
Part 7 of 16
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 8 of 16
Learning cycle 2 of 2
Part 2 · Venn diagrams
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 16
Learn
Venn diagrams
In a Venn diagram, the overlap holds members of both sets. Fill the intersection first, then subtract to complete each remaining region, and place anything left over outside the circles. If 18 study French, 14 study German and 6 study both, then 12 study only French and 8 study only German.
Explore the idea
Part 10 of 16
Learn
Set notation
A ∩ B means members of both sets; A ∪ B means members of at least one. From the language example, |A ∪ B| = 12 + 6 + 8 = 26. Notice that adding 18 and 14 gives 32, double-counting the six who take both.
Quick check
Part 11 of 16
Quick check
18 French, 14 German, 6 both. Only German is
Quick check
Part 12 of 16
Quick check
Why use a sample space grid?
Explore the idea
Part 13 of 16
Worked example
Worked answer: 18 study French, 14 German, 6 both, in a class of 30. How many study neither? (3 marks)
Six study both, so 18 − 6 = 12 study only French and 14 − 6 = 8 study only German (1). The number studying at least one language is 12 + 6 + 8 = 26 (1). Therefore 30 − 26 = 4 study neither (1). Adding 18 and 14 directly gives 32, which exceeds the class size and shows the double-counting error.
Challenge round
Part 14 of 16
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Mastery quiz
Part 15 of 16
Marked quiz
End of lesson quiz: Listing outcomes with tables, grids and Venn diagrams
4 questions, marked with the reasoning shown. No timer.
1. Rolling two dice, P(total 7) is
2. A ∩ B means
3. 18 French, 14 German, 6 both. Only German is
4. Why use a sample space grid?
Lesson round-up
Part 16 of 16
Lesson round-up
Ready when you are
Quiz score
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Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 16 · Watch & discover
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Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams