Sheet № 240 · Foundation + Higher · AQA · Edexcel · OCR
Sample Space Diagrams –
Sample space diagrams are a key probability tool tested on AQA, Edexcel and OCR GCSE papers at Foundation and Higher tier. They let you list every possible outcome when two events are combined — for example, rolling two dice or spinning two spinners — and then calculate probabilities by counting favourable outcomes. Getting the diagram ri
§Key definitions
Question:
Two fair six-sided dice are rolled and their scores are added together. Find the probability that the total is 7.
Answer:
P(total is 7) = 6/36 = 1/6.
Q1 (Foundation):
Two fair six-sided dice are rolled and the scores added. Find the probability that the total is less than 5.
Q2 (Foundation):
A fair coin is flipped and a fair four-sided dice (1–4) is rolled. List the sample space and find P(tail and number > 2).
Q3 (Higher):
Two fair six-sided dice are rolled and the scores are multiplied. Find the probability that the product is a square number.
§Formulas to memorise
Total number of outcomes = number of outcomes for event 1 × number of outcomes for event 2
P(event) = number of favourable outcomes ÷ total number of outcomes
List the outcomes — of each event along the top and down the side of a grid.
Fill in every cell — with the combined result (e.g. the sum, product or pair).
Count the total — number of outcomes (this should match the product rule).
Count the favourable outcomes — those that satisfy the condition in the question.
Calculate the probability — as favourable ÷ total and simplify.
Worked example
Two fair six-sided dice are rolled and their scores are added together. Find the probability that the total is 7.
Working:
⚠ Common mistakes
- ✗Missing outcomes. If you do not draw a complete grid, you risk leaving out combinations. Always check that the number of cells matches the product rule.
- ✗Double-counting. When listing pairs like (2,5) and (5,2), these are different outcomes for two dice — count them both.
- ✗Confusing sums and products. Read the question carefully to see whether you should add, multiply or simply list the pair.
✦ Exam tips
- →A well-drawn grid is quick and almost guarantees full marks — spend the time setting it up neatly.
- →For two dice, the total is always 36 outcomes. Memorise the symmetry: the most likely sum is 7 (six ways), with sums of 2 and 12 each having only one way.
- →If asked for P(at least 9), count outcomes that give 9, 10, 11 and 12.
- →For extensions using tree diagrams, see probability tree diagrams.