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Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M37 · Sample spaces, sets and Venn diagrams

Revision notes, worked examples and methods for sample spaces, sets and venn diagrams.

Notes and quizzes ready · 50 questions · Video coming soon.

Revise the key ideas

Sample spaces and systematic outcomes

  • A sample space lists all possible outcomes. For one coin toss it is {H, T}; for two tosses it is {HH, HT, TH, TT}. HT and TH are distinct ordered outcomes.
  • A table is useful when combining two experiments. Put one experiment's outcomes along each side, and fill every cell with the resulting pair or value.
    Sample-space table for sums of two diceRows and columns show die faces one to six. The thirty-six cells contain sums, with six sum-seven cells highlighted.Sum of two fair dice123456123456723456783456789456789105678910116789101112Six of the 36 outcome cells give sum 7
    Sample-space table for sums of two dice
    Enlarge diagram: Sample-space table for sums of two dice
  • Worked example: Two fair dice have 36 equally likely ordered pairs. Sum 7 occurs in 6 cells, so P(sum 7) = = .
  • Possible sums 2 to 12 are not equally likely. There is one way to make 2 and six ways to make 7, so counting the eleven sums equally would be wrong.
  • For a coin and a three-outcome spinner, a table has 2 × 3 = 6 cells. Equal probabilities in the cells require each device to have equally likely outcomes and independence.

Sets and Venn diagrams

  • A set is a collection of elements. The universal set, often written ξ, contains every element under consideration. A′ is the complement of A, containing elements in ξ but outside A.
  • A ∩ B means elements in both sets; A ∪ B means elements in A or B or both. “Or” in a union includes the overlap.
    French and Spanish Venn diagramThirteen students study only French, five both, seven only Spanish and five neither. Counts sum to thirty.FrenchSpanish13575Total 30; overlap means both languages
    French and Spanish Venn diagram
    Enlarge diagram: French and Spanish Venn diagram
  • Put elements belonging to both sets in the overlap first. Then fill A-only and B-only regions, and finally the region outside both circles.
  • For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The overlap is subtracted once because it was included in both totals.

Worked examples

  • Worked example: Of 30 students, 18 study French, 12 study Spanish and 5 study both. French only = 13, Spanish only = 7, either language = 25 and neither = 5.
  • For the same group, P(French and Spanish) = = and P(French or Spanish) = = . Both use the full group as denominator.
  • To find P(A′ ∩ B), use the B-only region, not everything outside A. Reading the symbols carefully avoids choosing the wrong region.
  • If a question uses three sets, work from the central triple overlap outwards and use the total to check the remaining region. Follow all given counts consistently.
  • Counts must be non-negative and sum to the stated total. A negative “only” region is a sign that the data were interpreted incorrectly.

Test yourself

50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.

Revision video

Video coming soon.