Year 9 · Mathematics · Probability
Venn diagrams and two-way tables
Organise outcomes using Venn diagrams and two-way tables, and use them to find probabilities involving "and", "or" and "not".
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- Organise data into Venn diagrams and two-way tables.
- Interpret the words “and”, “or” and “not” in probability.
- Calculate probabilities from Venn diagrams and two-way tables.
Success looks like
- I can complete a Venn diagram, including the overlap and the region outside both circles.
- I can complete the missing totals in a two-way table.
- I can find probabilities such as P(A and B), P(A or B) and P(not A).
The big idea
A Venn diagram shows how two groups overlap. A rectangle represents everyone (the sample space), and each circle represents one group. The overlap holds members of both groups, and the space outside both circles holds members of neither. Every person or outcome goes in exactly one region.
When filling in a Venn diagram, start with the overlap. If 12 students play soccer, 15 play tennis and 5 play both, put 5 in the overlap first. Then soccer only = 12 − 5 = 7 and tennis only = 15 − 5 = 10.
The words “and”, “or” and “not” have exact meanings in probability:
- A and B: in both groups — the overlap only.
- A or B: in A, in B, or in both — everything inside the circles.
- Not A: everything outside circle A, including outside both circles. P(not A) = 1 − P(A).
A two-way table shows the same kind of information in rows and columns. Each row and column has a total, and the grand total is in the bottom-right corner. Missing values can be found by adding or subtracting along a row or column.
To find a probability from either display, count the favourable outcomes and divide by the total: P(event) = number in the event ÷ total number. Be careful not to count the overlap twice when working out “or”.
Worked examples
Using a Venn diagram
In a class of 30, 12 students play soccer, 15 play tennis and 5 play both. Find P(soccer and tennis), P(soccer or tennis) and P(not soccer) for a student chosen at random.
- Overlap (both) = 5. Soccer only = 12 − 5 = 7. Tennis only = 15 − 5 = 10.
- Inside the circles: 7 + 5 + 10 = 22, so neither = 30 − 22 = 8.
- P(soccer and tennis) = 530 = 16.
- P(soccer or tennis) = 2230 = 1115.
- Not soccer = tennis only + neither = 10 + 8 = 18, so P(not soccer) = 1830 = 35.
Answer: P(S and T) = 16, P(S or T) = 1115, P(not S) = 35
Completing a two-way table
Of 40 people, 8 are left-handed. 3 of the left-handed people and 12 of the right-handed people wear glasses. Complete a two-way table.
- Right-handed = 40 − 8 = 32.
- Left-handed without glasses = 8 − 3 = 5. Right-handed without glasses = 32 − 12 = 20.
- Glasses total = 3 + 12 = 15. No glasses total = 5 + 20 = 25. Check: 15 + 25 = 40.
Answer: Left: 3 glasses, 5 no glasses (8). Right: 12 glasses, 20 no glasses (32). Totals: 15 glasses, 25 no glasses, 40 people.
Probabilities from the table
Using the table above, find P(left-handed and wears glasses), P(left-handed or wears glasses) and P(no glasses).
- Left-handed and glasses = 3, so P = 340.
- Left-handed or glasses: count each cell once — 3 (left, glasses) + 5 (left, no glasses) + 12 (right, glasses) = 20, so P = 2040 = 12.
- No glasses = 25, so P = 2540 = 58.
Answer: 340, 12 and 58
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
In a group of 50 people, 28 like cats, 20 like dogs and 8 like both. How many like cats only?
20 — Cats only = 28 − 8 = 20.
In the same group of 50 (28 like cats, 20 like dogs, 8 like both), how many like neither?
10 — Cats only 20 + both 8 + dogs only 12 = 40 inside the circles, so 50 − 40 = 10 like neither.
In the same group, what is the probability that a person chosen at random likes both cats and dogs?
425 — 8 of the 50 people like both: 850 = 425.
In a Venn diagram, the event “A or B” means:
everything inside A, B or both — “Or” includes A only, B only and the overlap.
In the same group of 50 (28 like cats, 20 like dogs, 8 like both), find P(likes cats or dogs) as a fraction in simplest form.
4/5 — 40 people are inside the circles: 4050 = 45.
In the same group of 50, find P(does not like dogs) as a fraction in simplest form.
3/5 — 50 − 20 = 30 people do not like dogs: 3050 = 35.
A two-way table of 60 students shows: owns a bike and has a pet 18; owns a bike, no pet 12; no bike, has a pet 22; no bike, no pet 8. Find P(owns a bike and has a pet) as a fraction in simplest form.
3/10 — 1860 = 310.
Using the same table of 60 students, find P(owns a bike or has a pet) as a fraction in simplest form.
13/15 — Count each cell once: 18 + 12 + 22 = 52. 5260 = 1315. (Only the 8 with no bike and no pet are left out.)
Watch out for
- Putting the full group totals into the circles, so the overlap is counted twice (for example writing 28 in the cats-only region). Always fill the overlap first, then subtract.
- Treating “or” as “one or the other but not both”. In maths, “A or B” includes the overlap.
- Forgetting the region outside both circles. Check that all the regions add up to the total.
Tips for parents
- Make a floor Venn diagram with two hoops and sort family members or toys by two features, such as “wears glasses” and “has brown hair”.
- Always ask the learner to check that the numbers in all regions (or all table cells) add to the total.
- Read “and”, “or” and “not” questions aloud and have the learner shade the matching region before counting.
Go further
- Survey family and friends about two yes/no questions, then display the results as both a Venn diagram and a two-way table.
- Show the same data as a Venn diagram and a two-way table, and match each table cell to a Venn region.
- Investigate the addition rule: check that P(A or B) = P(A) + P(B) − P(A and B) using the worked example numbers.
About 50 minutes
You'll need
- Paper, pencil and ruler
- Two hoops or loops of string (to make a floor Venn diagram)
- Small objects or cards to sort
Key words
- Venn diagram
- A diagram that uses overlapping circles inside a rectangle to show how groups share members.
- Two-way table
- A table that sorts data by two features at once, with row and column totals.
- A and B
- Outcomes that are in both A and B — the overlap of the circles.
- A or B
- Outcomes in A, in B, or in both — everything inside either circle.
- Not A (complement)
- Every outcome that is not in A. P(not A) = 1 − P(A).
- Mutually exclusive
- Events that cannot happen at the same time, so their circles do not overlap.
Still tricky?
For parents
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