Venn Diagrams and Two-Way Tables
Theory
A Venn diagram and a two-way table both sort a group by two attributes at once. This Year 9 maths page shows how to fill in the regions or cells, read "and", inclusive "or", exclusive "or" and "not", spot mutually exclusive events, and find probabilities and percentages from either display.
A Venn diagram shows a whole group as a rectangle, called the universal set \(\xi\). Each attribute is a circle. Where circles overlap, people have both attributes.
With two circles \(A\) and \(B\) there are four regions: \(A\) only, both, \(B\) only, and neither. Every member of the group is in exactly one region, so the four counts add to the total.
The intersection \(A \cap B\) is the overlap: in \(A\) and in \(B\). The union \(A \cup B\) is everything inside either circle: in \(A\) or \(B\) or both. The complement \(A'\) is everything outside \(A\). \(n(A)\) means the number in \(A\).
Events are mutually exclusive if they cannot happen together, like rolling a \(2\) and rolling a \(5\) with one die. Their circles do not overlap.
A two-way table records the same kind of information in rows and columns. Each inner cell is an and, such as boys who walk. A row total or column total is one attribute on its own, and the grand total is the whole group.
Inclusive or means one or the other or both: everything inside the two circles. Exclusive or means one or the other but not both: the two "only" regions. Questions often say "exactly one" or "but not both" for the exclusive or.
Probability from a Venn diagram:
The overlap is counted in both circle totals, so:
For a two-way table, an or is a row plus a column, less the cell they share:
How to solve a Venn diagram problem
- Draw the rectangle and the circles, and label them.
- Fill the overlap first. If it is unknown, find it: add the circle totals and subtract the number in at least one.
- Subtract the overlap from each circle total to get the "only" regions.
- Find the outside region: total minus everything in the circles.
- Read the region the question asks for, and divide by the whole group for a probability. Check the regions add to the total.
How to use a two-way table
- Complete the table first: add a Total row and a Total column, and fill any gaps by subtraction.
- Check the grand total three ways: cells, row totals, column totals.
- Decide what the question wants: one cell (and), a row or column (one attribute), or a row plus a column (or).
- Count, taking off the shared cell for an or.
- Divide by the right total: the grand total for a probability of the whole group, or a row or column total for a percentage of that group.
Put the overlap in first. Then take it out of each circle total.
| \(\text{chess only}\) | \(=\) | \(15 - 6 = 9\) |
| \(\text{cards only}\) | \(=\) | \(11 - 6 = 5\) |
| \(\text{in a circle}\) | \(=\) | \(9 + 6 + 5 = 20\) |
| \(\text{neither}\) | \(=\) | \(25 - 20 = \mathbf{5}\) |
"Exactly one" means the two "only" regions, not the overlap.
| \(\text{swim only}\) | \(=\) | \(25 - 8 = 17\) |
| \(\text{run only}\) | \(=\) | \(19 - 8 = 11\) |
| \(P(\text{exactly one})\) | \(=\) | \(\dfrac{17 + 11}{44} = \dfrac{28}{44}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{7}{11}}\) |
(i) is a girl who walks
(ii) walks.
| Walks | Does not walk | Total | |
|---|---|---|---|
| Boys | 18 | 12 | 30 |
| Girls | 21 | 14 | 35 |
| Total | 39 | 26 | 65 |
| Walks | Does not walk | Total | |
|---|---|---|---|
| Boys | 18 | 12 | 30 |
| Girls | 21 | 14 | 35 |
| Total | 39 | 26 | 65 |
(i) "Girl and walks" is one cell. Divide it by the grand total.
| \(P(\text{girl and walks})\) | \(=\) | \(\mathbf{\dfrac{21}{65}}\) |
It does not simplify, because \(21 = 3 \times 7\) and \(65 = 5 \times 13\).
(ii) "Walks" says nothing about gender, so use the whole Walks column.
| \(\text{walk}\) | \(=\) | \(18 + 21 = 39\) |
| \(P(\text{walks})\) | \(=\) | \(\dfrac{39}{65} = \mathbf{\dfrac{3}{5}}\) |
| Walks | Does not walk | Total | |
|---|---|---|---|
| Boys | 18 | 12 | 30 |
| Girls | 21 | 14 | 35 |
| Total | 39 | 26 | 65 |
Add the Boys row total and the Does-not-walk column total. The boys who do not walk are in both, so take them off once.
| \(n(\text{boys or not walk})\) | \(=\) | \(30 + 26 - 12\) |
| \(\) | \(=\) | \(\mathbf{44}\) |
For a percentage of the boys, divide by the boys' total, not by \(65\).
| \(\dfrac{12}{30} \times 100\%\) | \(=\) | \(\mathbf{40\%}\) |
Common pitfalls
Frequently asked questions
How do you fill in a Venn diagram?
Start with the overlap. Subtract it from each circle total to get the "only" regions. Then subtract everything in the circles from the total to get the region outside.
What are mutually exclusive events?
Events that cannot happen at the same time, such as rolling a \(2\) and rolling a \(5\) on one die. On a Venn diagram their circles do not overlap.
What is the difference between inclusive or and exclusive or?
Inclusive or means one or the other or both, so it includes the overlap. Exclusive or means one or the other but not both, so it is only the two "only" regions. In Example 2, swimming or running (inclusive) is \(17 + 8 + 11 = 36\) people, but exactly one of them (exclusive) is \(28\).
How do you find probability from a two-way table?
Count the people who fit the event, from a cell, a row, a column or a combination, and divide by the grand total.
How is a two-way table related to a Venn diagram?
They show the same information. The four cells of a two-by-two table match the four regions of a two-circle Venn diagram: both, one only, the other only, and neither.