Venn Diagrams
A Venn diagram sorts a group by two attributes: each circle holds one attribute, the overlap holds both and the region outside holds neither. This NSW Year 8 maths subtopic goes beyond Stage 4, since the NSW syllabus places Venn diagrams in Stage 5, and it uses the Stage 4 rule of favourable outcomes over the total.
You will read and complete diagrams from lists and from totals, find a missing overlap by adding the two circles and taking away the “or” total, and recognise mutually exclusive events, whose circles do not overlap. You will then find probabilities for “and”, “or”, “only” and “neither” from counts such as students in the choir and band, or hikers who saw a kangaroo or a wombat. It is part of Year 8 Probability.
Theory
A Venn diagram sorts one group by two attributes so that every member sits in exactly one region: A only, both, B only, or neither. With netball only \(8\), both \(3\), hockey only \(7\) and neither \(2\), there are \(20\) people and \(P(\text{both}) = \dfrac{3}{20}\). This Year 8 maths page shows how.
The picture. The rectangle holds everyone in the group. Each circle holds the members with one attribute, and the overlap holds those with both. Write a count in every region, including the one outside the circles. The four counts add to the total.
From a list. Put each member in the one region that matches both answers, then count.
From totals. A circle total includes the overlap, so fill the overlap first, then subtract: A only \(= n(\text{A}) - n(\text{both})\), and neither \(=\) total \(-\) everyone inside the circles.
A missing overlap. Adding the two circle totals counts the overlap twice, so \(n(\text{both}) = n(\text{A}) + n(\text{B}) - n(\text{A or B})\).
Mutually exclusive events cannot happen together: the overlap is empty, the circles are drawn apart, and “or” is just the two circles added.
Probability. Count the region, then divide by the total: \(P = \dfrac{\text{count in the region}}{\text{total}}\).
Which regions to add:
| In words | Add these regions |
|---|---|
| A and B (both) | the overlap |
| A only | A circle without the overlap |
| A | A only + both |
| A or B | A only + both + B only |
| exactly one | A only + B only |
| not B | A only + neither |
| neither | outside both circles |
| the total | all four regions |
Building a Venn diagram from totals
- Write the overlap (both) first.
- Subtract it from each circle total to get the “only” regions.
- Subtract everyone inside from the total to get neither.
- Check that the four regions add to the total.
(i) only netball
(ii) netball and hockey
(iii) neither.
Netball: \(8 + 3 = 11\). Hockey: \(3 + 7 = 10\). Total: \(8 + 3 + 7 + 2 = 20\).
| \(P(\text{only netball})\) | \(=\) | \(\dfrac{8}{20} = \dfrac{2}{5}\) |
| \(P(\text{both})\) | \(=\) | \(\dfrac{3}{20}\) |
| \(P(\text{neither})\) | \(=\) | \(\dfrac{2}{20} = \dfrac{1}{10}\) |
(i) \(\dfrac{2}{5}\)
(ii) \(\dfrac{3}{20}\)
(iii) \(\dfrac{1}{10}\)
(i) the number who play both
(ii) the number in the guitar circle
(iii) the number who play neither.
(i) \(4\) (the overlap)
(ii) \(11 + 4 = 15\)
(iii) \(7\) (outside the circles)
| \(\text{Athletics only}\) | \(=\) | \(16 - 6 = 10\) |
| \(\text{Debating only}\) | \(=\) | \(13 - 6 = 7\) |
| \(\text{Neither}\) | \(=\) | \(30 - 23 = 7\) |
| \(P(\text{exactly one})\) | \(=\) | \(\dfrac{17}{30}\) |
\(P(\text{exactly one}) = \dfrac{10 + 7}{30} = \dfrac{17}{30}\).
Both: \(2\) (Ana, Dee). I only: \(2\) (Ben, Fay). S only: \(3\) (Cy, Gus, Hal). Neither: \(1\) (Eli). Check: \(2 + 2 + 3 + 1 = 8\).
(i) \(P(\text{both})\)
(ii) \(P(\text{science or art})\)
(iii) \(P(\text{neither})\) as a percentage.
The total is \(18 + 9 + 15 + 8 = 50\), and \(18 + 9 + 15 = 42\) are inside the circles.
| \(P(\text{either})\) | \(=\) | \(\dfrac{42}{50} = \dfrac{21}{25}\) |
| \(P(\text{neither})\) | \(=\) | \(\dfrac{8}{50} = \dfrac{16}{100}\) |
(i) \(\dfrac{9}{50}\)
(ii) \(\dfrac{21}{25}\)
(iii) \(16\%\)
(i) how many enter both
(ii) \(P(\text{summer only})\).
| \(\text{Inside}\) | \(=\) | \(48 - 5 = 43\) |
| \(n(\text{both})\) | \(=\) | \(30 + 26 - 43 = 13\) |
| \(\text{Summer only}\) | \(=\) | \(30 - 13 = 17\) |
(i) \(13\)
(ii) \(\dfrac{17}{48}\). Check: \(17 + 13 + 13 + 5 = 48\).
Common pitfalls
Frequently asked questions
What does the overlap in a Venn diagram mean?
The members who have both attributes.
How do I find the number in A or B?
Add the three regions inside the circles: A only, both and B only.
What are mutually exclusive events?
Events that cannot happen together, so their circles do not overlap.
How do I find the overlap from totals?
Add the two circle totals and subtract the number in A or B: \(n(\text{both}) = n(\text{A}) + n(\text{B}) - n(\text{A or B})\).
How do I find a probability from a Venn diagram?
Count the members in the region you want and divide by the total in all four regions.