M37 · Sample spaces, sets and Venn diagrams | Edexcel GCSE Maths

Revise the key ideas

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 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) = 636 = 16.
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.

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 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 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) = 530 = 16 and P(French or Spanish) = 2530 = 56. 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.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.