Venn Diagrams
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.