Enumerating outcomes with tables, grids and Venn diagrams

WJEC Unit 2 has at least one Venn diagram or two-way table question every paper, worth 4–6 marks.

Sample-space tables

For two events with small outcome counts, draw a grid.

Two fair dice, sum of scores:

| + | 1 | 2 | 3 | 4 | 5 | 6 | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | … and so on |

Total cells = 36. Count favourable outcomes, divide by 36.

• P(sum = 7) = 6/36 = 1/6.

Two-way tables

Used for actual frequency data.

| | Bus | Walk | Total | | Year 10 | 18 | 22 | 40 | | Year 11 | 12 | 28 | 40 | | Total | 30 | 50 | 80 |

P(year 10 student walks) = 22/40 = 11/20. P(student is year 11 AND took bus) = 12/80 = 3/20.

Venn diagrams — set notation

• ξ (or U) — universal set (all elements).
• A ∪ B — A union B (in A or B or both).
• A ∩ B — A intersection B (in BOTH A and B).
• A' — complement of A (not in A).
• ∅ — empty set.
• nA — number of elements in A.

Filling a Venn diagram with frequencies

Always start from the innermost region (intersection) and work outwards.

Example: 50 students. 30 study French (F), 25 study Spanish (S), 12 study both.

1. F ∩ S = 12 (centre).
2. F only = 30 − 12 = 18.
3. S only = 25 − 12 = 13.
4. Neither = 50 − (18 + 12 + 13) = 50 − 43 = 7 (outside both circles, inside ξ).

Probability from a Venn diagram

P(F) = 30/50 = 3/5. P(F ∩ S) = 12/50 = 6/25. P(F ∪ S) = (18 + 12 + 13)/50 = 43/50. P(F' ∩ S') = 7/50.

WJEC exam tip

When filling a Venn diagram with three or more sets, start with the centre intersection and work outwards. Filling outer regions first produces double-counting errors and loses M-marks.