2 way tables
A two-way table records a group by two attributes at once, one in the rows and the other in the columns, with row, column and grand totals. This NSW Year 8 maths subtopic goes beyond Stage 4, since the NSW syllabus places two-way tables in Stage 5, and it uses the Stage 4 probability rule.
You will fill in missing cells by subtraction, build a table from a description and convert between a two-way table and a Venn diagram. You will find probabilities with the grand total underneath, including “and”, “or” and “neither” events, and spot the mistake of dividing by a row total. Canteen sales, excursion choices, beachgoers and footy supporters wearing scarves are typical examples. It is part of Year 8 Probability.
Theory
A two-way table sorts one group by two attributes at once and shows the totals of every row and column. Each person is in exactly one inner cell, and a probability is a count over the grand total: if \(3\) of \(31\) students are boys who cannot swim, \(P = \dfrac{3}{31}\). This Year 8 maths page shows how.
Rows and columns. One attribute gives the rows and the other gives the columns. Every person is counted in exactly one of the inner cells.
The totals. Each row adds to its row total and each column to its column total. The bottom-right cell is the grand total: add the row totals or the column totals; doing both is the check.
A missing number is found by subtraction: if a row reads \(5\), \(?\) and total \(14\), the missing cell is \(14 - 5 = 9\).
Probability: \(P(\text{event}) = \dfrac{\text{count from the table}}{\text{grand total}}\).
‘And’ is a single inner cell. ‘Or’ is the row total plus the column total, minus the shared cell once. ‘Neither’ is the inner cell with both “no” labels.
Tables and Venn diagrams hold the same four counts: each inner cell is one region of the Venn diagram.
The students of a class sorted by pets:
| Owns a cat | No cat | Total | |
|---|---|---|---|
| Owns a dog | 6 | 13 | 19 |
| No dog | 5 | 9 | 14 |
| Total | 11 | 22 | 33 |
\(P(\text{dog and cat}) = \dfrac{6}{33} = \dfrac{2}{11}\), \(P(\text{dog or cat}) = \dfrac{19 + 11 - 6}{33} = \dfrac{8}{11}\) and \(P(\text{neither}) = \dfrac{9}{33} = \dfrac{3}{11}\).
Completing a two-way table
- Fill any cell whose row or column has only one gap, by subtraction.
- Repeat until every cell is full.
- Check the grand total across the rows and down the columns.
Probability from a table
- Find the count: one cell for “and”; row \(+\) column \(-\) shared cell for “or”.
- Divide by the grand total and simplify.
(i) How many boys are there?
(ii) How many students can swim?
(iii) How many girls cannot swim?
(iv) Find the probability of randomly selecting a boy who cannot swim.
| Can swim | Cannot swim | Total | |
|---|---|---|---|
| Boys | 12 | 3 | 15 |
| Girls | 14 | 2 | 16 |
| Total | 26 | 5 | 31 |
| \(\text{Boys}\) | \(=\) | \(12 + 3 = 15\) |
| \(\text{Can swim}\) | \(=\) | \(12 + 14 = 26\) |
| \(P\) | \(=\) | \(\dfrac{3}{31}\) |
(i) \(15\)
(ii) \(26\)
(iii) \(2\)
(iv) \(\dfrac{3}{31}\)
| \(\text{8A}\) | \(=\) | \(27 + 18 = 45\) |
| \(\text{8B}\) | \(=\) | \(22 + 23 = 45\) |
| \(\text{Kayak}\) | \(=\) | \(27 + 22 = 49\) |
| \(\text{Walk}\) | \(=\) | \(18 + 23 = 41\) |
Grand total \(90\) both ways: \(45 + 45 = 90\) and \(49 + 41 = 90\).
| Bin | No bin | Total | |
|---|---|---|---|
| Recycle | 40 | 56 | 96 |
| Don’t | 28 | 36 | 64 |
| Total | 68 | 92 | 160 |
| \(\text{Recycle only}\) | \(=\) | \(96 - 40 = 56\) |
| \(\text{Bin only}\) | \(=\) | \(68 - 40 = 28\) |
| \(\text{Neither}\) | \(=\) | \(160 - 124 = 36\) |
\(P(\text{neither}) = \dfrac{36}{160} = \dfrac{9}{40}\)
| 8M | Choir | No choir | Total |
|---|---|---|---|
| Band | 8 | 15 | 23 |
| No band | 9 | 4 | 13 |
| Total | 17 | 19 | 36 |
| \(36 - 4\) | \(=\) | \(32\) |
| \(23 + 17 - 8\) | \(=\) | \(32\) |
| \(P\) | \(=\) | \(\dfrac{32}{36} = \dfrac{8}{9}\) |
Each inner cell is one region. \(P(\text{band or choir}) = \dfrac{8}{9}\).
(i) \(P(\text{senior and } 100 \text{ m})\) as a percentage
(ii) \(P(\text{junior or relay})\) as a decimal.
| \(\dfrac{35}{250}\) | \(=\) | \(\dfrac{14}{100} = 14\%\) |
| \(160 + 100 - 65\) | \(=\) | \(195\) |
| \(\dfrac{195}{250}\) | \(=\) | \(0.78\) |
(i) \(14\%\)
(ii) \(0.78\) (the \(65\) junior relay entries are counted once)
Common pitfalls
Frequently asked questions
What is a two-way table?
A table that sorts one group by two attributes at once, with totals for every row and column.
How do I find a missing value?
Subtract the known cells from the row or column total.
How do I find a probability from a two-way table?
Take the count you need and divide by the grand total.
How do I find the probability of A or B?
Add the row total and the column total, subtract the cell they share, then divide by the grand total.
How is a two-way table related to a Venn diagram?
They show the same four counts: each inner cell of the table is one region of the Venn diagram.