Tables, sample-space grids and Venn diagrams
When the sample space gets complex, you need a visual way to organise outcomes. The three workhorses are tables, grids and Venn diagrams.
Sample-space grids (for two combined experiments)
For two experiments — most often two dice — set up a 2D grid:
- Rows = outcomes of experiment 1.
- Columns = outcomes of experiment 2.
- Cells = combined outcome.
For two fair 6-sided dice, the grid has 6 × 6 = 36 cells, all equally likely.
Examples on the grid:
- "Sum is 7": cells where row + col = 7. There are 6 such cells (1+6, 2+5, …, 6+1). P = 6/36 = 1/6.
- "Both even": rows 2, 4, 6 paired with columns 2, 4, 6 → 9 cells. P = 9/36 = 1/4.
- "Difference is 2": (1,3),(2,4),(3,5),(4,6) and reverses → 8 cells. P = 8/36 = 2/9.
The grid lets you count visually and avoid double-counting.
Two-way tables (for categorical data)
We met these in P1. Use them whenever you need to handle two categorical variables — gender × subject, year × travel mode, etc. Read marginal totals from the row/column sums.
Venn diagrams
Venn diagrams partition the sample space using overlapping circles, one per event.
For two events A and B:
- A only.
- A and B (the overlap).
- B only.
- Outside both (the universal set minus A∪B).
For three events you have 8 regions (2³), including the centre A∩B∩C and a triple-only region.
Filling a 2-event Venn from data
Always start with the intersection (the overlap), then subtract to fill the "only" regions.
Worked example: in a class of 30, 18 study French, 14 study Spanish, 6 study both.
- A∩B (French and Spanish) = 6.
- A only (French only) = 18 − 6 = 12.
- B only (Spanish only) = 14 − 6 = 8.
- Outside = 30 − (12 + 6 + 8) = 4.
P(French | Spanish) = 6/14 = 3/7.
Set notation (for Higher tier)
- A ∪ B: union — in A or B (or both).
- A ∩ B: intersection — in both A AND B.
- A': complement — not in A.
- ξ: universal set.
- |A|: number of elements in A.
Probabilities from the diagram
Once a Venn is filled with counts (or probabilities), reading off probabilities is direct.
PA = (count in A) / (universal total). P(A ∩ B) = (count in overlap) / (universal total). P(A | B) = (count in overlap) / (count in B).
⚠Common mistakes— Common mistakes (examiner traps)
- Filling a Venn diagram outside-in. Always fill the central overlap first.
- Double-counting when you forget that the "A" circle includes A∩B.
- Missing the outside region. Anyone not in A or B still counts toward the universal total.
- Confusing P(A | B) with P(A ∩ B). Conditional divides by |B|; intersection divides by total.
- Forgetting a sample-space grid is symmetric. (1,3) and (3,1) are different cells.
➜Try this— Quick check
In a school of 100 students, 60 play football, 40 play basketball, and 25 play both. Draw a 2-event Venn diagram and find: (a) P(football only). (b) P(neither).
- F only = 60 − 25 = 35; B only = 40 − 25 = 15; both = 25; neither = 100 − 75 = 25.
- (a) 35/100 = 7/20; (b) 25/100 = 1/4.
AI-generated · claude-opus-4-7 · v3-deep-probability