Sample spaces, possibility tables and Venn diagrams

Sample spaces and possibility tables

A sample space is the set of all possible outcomes of an experiment.

Possibility table (or two-way table): a grid showing all outcomes of two combined events. Useful for rolling dice, spinning spinners, or any two-stage experiment with manageable outcome counts.

Example: two dice, each numbered 1–6. Total outcomes = 36. The table has rows 1–6 (die 1) and columns 1–6 (die 2). Each cell is a sum or product.

• P(sum = 7) = 6/36 = 1/6 (six ways: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1).

Venn diagrams

A Venn diagram shows the overlap between sets. Circles represent sets; the intersection (overlap) shows elements that belong to both.

Notation:

• A ∪ B (union): elements in A or B (or both).
• A ∩ B (intersection): elements in A and B.
• A' (complement): elements NOT in A.
• ξ (universal set / "everything"): the rectangle containing all circles.

Filling in a Venn diagram:

1. Start with the intersection (elements in both sets).
2. Work outwards to each circle only.
3. Any elements outside both circles go in the rectangle.

Example: 30 students. 18 study French (F), 12 study Spanish (S), 7 study both. F only = 18 − 7 = 11. S only = 12 − 7 = 5. Neither = 30 − 11 − 7 − 5 = 7.

Probability from Venn diagrams:

• P(F ∪ S) = (11 + 7 + 5)/30 = 23/30.
• P(F ∩ S) = 7/30.
• P(F only) = 11/30.
• P(neither) = 7/30.

Three-set Venn diagrams (Higher)

Three overlapping circles. Fill from the innermost intersection outward:

1. Intersection of all three sets.
2. Each pair's intersection (minus all-three).
3. Each set only.
4. The outside region.

CCEA context

CCEA frequently combines Venn diagrams with probability questions: fill in the diagram, then find probabilities including conditional probability (find P(A|B) — given B has occurred, what is PA?). Paper 2 context: student surveys, set membership.