Venn Diagrams & Two-Way Tables
Master Venn diagrams and two-way tables for NSW Year 12 Mathematics Standard 2. In this topic you sort a group by two attributes into four regions — only A, both, only B and neither — and read counts and probabilities straight off the display.
You will learn to construct and interpret a two-attribute Venn diagram and two-way table, use the addition rule to find “both” or “at least one”, and convert between the two representations — a core Standard 2 probability skill for surveys and real-world data.
Theory
A Venn diagram and a two-way table organise data sorted by two attributes. This Year 12 Standard 2 (NSW) guide shows how to read counts and probabilities from each, use the addition rule for “at least one”, and convert a Venn diagram into a two-way table.
A Venn diagram uses two overlapping circles to sort a group by two yes/no attributes. The overlap holds the items with both attributes, and the space outside both circles holds those with neither.
A two-way table shows the same information in rows and columns: one attribute across the top, the other down the side, with row, column and grand totals in the margins.
The four groups — only A, both, only B and neither — never overlap and always add up to the total. Because both displays hold the same counts you can convert freely between them, and read a probability as a region count divided by the total.
| Swimming | No swimming | Total | |
|---|---|---|---|
| Cricket | 7 | 12 | 19 |
| No cricket | 6 | 5 | 11 |
| Total | 13 | 17 | 30 |
Read a probability as a region count over the total:
The addition rule (inclusion–exclusion) counts each item once, so the overlap is subtracted:
How to build and read the displays
- Identify the two attributes and the group total.
- Fill the overlap first (the both count), then each “only” region, then the “neither” region outside both circles.
- Check that all four regions add up to the total.
- Read counts: an attribute total is its “only” region plus the overlap; “at least one” is total minus neither.
- Probability: divide the required region count by the total. To convert to a two-way table, put the both count in the corner cell and the attribute totals in the margins.
A painting count includes the overlap; “neither” sits outside both circles.
| \(n(\text{Painting})\) | \(=\) | \(9+6\) |
| \(\) | \(=\) | \(15\) |
| \(n(\text{Neither})\) | \(=\) | \(5\) |
Each row and each column must add to its total.
| Licence | No licence | Total | |
|---|---|---|---|
| Passport | 28 | ? | 34 |
| No passport | ? | 4 | 16 |
| Total | ? | 10 | 50 |
| \(\text{Passport, no licence}\) | \(=\) | \(34-28=6\) |
| \(\text{Licence total}\) | \(=\) | \(50-10=40\) |
So 6 hold a passport but no licence, and 40 hold a licence in total.
The overlap is the “both” cell; the margins are the attribute totals.
| \(\text{Weights}\) | \(=\) | \(13+8=21\) |
| \(\text{Cardio}\) | \(=\) | \(8+9=17\) |
| \(\text{No weights}\) | \(=\) | \(9+5=14\) |
| Cardio | No cardio | Total | |
|---|---|---|---|
| Weights | 8 | 13 | 21 |
| No weights | 9 | 5 | 14 |
| Total | 17 | 18 | 35 |
Place the 12 who saw both in the overlap, then work outwards.
| \(\text{At least one}\) | \(=\) | \(38+27-12=53\) |
| \(\text{Neither}\) | \(=\) | \(60-53=7\) |
| \(P(\text{at least one})\) | \(=\) | \(\dfrac{53}{60}\) |
Common pitfalls
Frequently asked questions
What is the difference between a Venn diagram and a two-way table?
They show the same information in different shapes. A Venn diagram uses overlapping circles, with the overlap for items that have both attributes. A two-way table uses rows and columns with totals in the margins. You can always convert one into the other.
Does the number who like A include the people who like both?
Yes. When a question says a certain number play cricket, that count includes the people who play both cricket and something else. The only-cricket region is that total minus the overlap in the middle.
How do you find how many have both when you only know the totals?
Use the addition rule. The number with at least one attribute equals total minus neither. Then both equals n(A) plus n(B) minus the number with at least one. For example, 38 plus 27 minus 53 equals 12.
How do you convert a Venn diagram into a two-way table?
Put the overlap count in the corner cell (both attributes), put each only-region in the matching cell, and put the neither count in the no-and-no cell. Adding across and down then gives the row, column and grand totals.
How do you find a probability from a two-way table or Venn diagram?
Divide the count in the region you want by the grand total. For at least one attribute, take total minus neither over the total. For not A, use 1 minus the probability of A.
What does 'at least one' mean?
At least one means the item has the first attribute, the second attribute, or both — everyone except the neither group. So it equals the total minus the number in the neither region.