Summary — What This Chapter Covers
- Two-way tables: compare two categories (e.g. Year group × Language) — fill in missing numbers using row/column totals, then read off probabilities.
- Possibility (sample space) diagrams: grids that list every possible combined outcome of two events (like rolling two dice) — count what you want out of the total grid.
- Venn diagrams for probability: circles show overlapping categories (e.g. Spanish & German) — add up the right regions, divide by the right total.
- The golden rule everywhere: only count outcomes if they're equally likely. You can't just "count" the lottery.
- Some questions restrict you to a smaller group (e.g. "given the child is in Class B…") — then your denominator shrinks to just that group's total, not the whole population.
1. Two-Way Tables
What is a two-way table?
A two-way table is exactly what it sounds like: a table that sorts data by two characteristics at once. Think of it like a spreadsheet where the rows are one category and the columns are another category, and each cell tells you how many things/people fall into that specific combination.
Example: a college has 55 students. They can be split by year group (Year 12 or Year 13) AND by language (Spanish or German). A two-way table lets you see both splits in one picture:
| Spanish | German |
|---|
| Year 12 | 15 | 10 |
| Year 13 | 5 | 25 |
Notice this raw table has no totals yet — that's step one you need to add yourself.
Step 1: Always add the totals
Before you can answer any probability question, add a Total row at the bottom and a Total column on the right. The bottom-right corner cell is the overall total — and it must equal both the sum of the row totals AND the sum of the column totals. This is your built-in check that you haven't made an arithmetic slip.
| Spanish | German | Total |
|---|
| Year 12 | 15 | 10 | 25 |
| Year 13 | 5 | 25 | 30 |
| Total | 20 | 35 | 55 |
Examiner Tip
Always check your row totals add up to the overall total, and your column totals add up to the overall total too. If they don't match, every probability you calculate afterwards will be wrong — so catch the mistake here, before you even start on part (a)!
Step 2: Reading off probabilities
Once your table is complete, probability questions become simple division. The key skill is figuring out what the denominator (bottom number of the fraction) should be:
- If you're picking from the whole population, the denominator is the overall total (here, 55).
- If you're picking from one specific category only (e.g. "a student is picked from Year 13"), the denominator is just that category's total, not the whole grand total.
Examples from the table above:
- P(a random student studies Spanish AND is in Year 12) = 15/55
Worked example: Building a table from a word problem