Mathematics 7–10 · Year 10
Two-way tables from a class survey: conditional probability in both directions
Statistics and probability
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The idea
A two-way table of two categorical variables gives conditional probabilities by reading along a row or down a column, and P(A given B) is not P(B given A).
What you need
- A class survey of two categorical variables that raise no privacy concern (for example writing hand and whether the learner walked to school today)
- Whiteboard two-way table with row and column totals
How to do it
- Collect the two categorical answers from each learner by a show of hands into the four cells; fill row, column and grand totals.
- Compute the probability that a learner chosen at random is left-handed, then the probability given they walked, then the probability they walked given they are left-handed.
- Write each conditional probability as cell over row total or cell over column total and explain which total is the denominator and why.
- Decide whether the two variables look independent by comparing P(left) with P(left given walked).
- Extension: convert the table to a Venn diagram and a tree diagram and check the same probabilities appear.
What you should see
As a worked check, a table with 25 learners of whom 12 walked and 3 are left-handed, 2 of the left-handers among the walkers, gives P(left) = 3/25 = 0.12, P(left given walked) = 2/12 = 0.167 and P(walked given left) = 2/3 = 0.667; the two conditionals differ because their denominators are the row total and the column total. The class's own table gives its own values, computed the same way. The learner knows it worked when the two conditional probabilities are read from different totals and the four cell probabilities sum to 1.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- P(A given B) equals P(B given A).
- Given means and.
- A small difference between P(A) and P(A given B) proves the variables are linked.
Safety card
Hazards
No hazard is listed.
Controls
- choose survey questions that are not sensitive and keep counts, not names
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Probability B (Path); page read 2026-09-22MA5-PRO-P-01
- Australian Curriculum v9AC9M10P01AC9M10ST04
Sources
The pages the author read to write this activity.