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
- 20 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 20
Visual introductionPicture this
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.
In a nutshell
Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams
Learning cycle
Part 2 of 20
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 20
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.
Reset break
Part 4 of 20
Pause
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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.
Explore the idea
Part 5 of 20
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 6 of 20
Quick check
Part 7 of 20
Reset break
Part 8 of 20
Pause
That's sitting 2 of 5 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 9 of 20
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 10 of 20
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 11 of 20
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.
Reset break
Part 12 of 20
Pause
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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.
Quick check
Part 13 of 20
Quick check
Part 14 of 20
Explore the idea
Part 15 of 20
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.
Reset break
Part 16 of 20
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 20
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
Challenge round
Part 18 of 20
Game · Recall cards
Rolling two fair dice, how many equally likely outcomes are there in total?
Card 1 of 4
Mastery quiz
Part 19 of 20
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 fair dice, how many equally likely outcomes are there in total?
2. How many of those 36 outcomes give a sum of 7?
3. In a Venn diagram, where do items that satisfy both conditions go?
4. Rolling two dice, P(total 7) is
Lesson round-up
Part 20 of 20
Lesson round-up
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Enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams