Probability
Choose your practice ↓Probability and statistics together make up 15% of the paper on both tiers, and probability is the half that depends most on setting the work out clearly. It begins with recording outcomes in tables and frequency trees, the idea of fairness and equally likely outcomes, relative frequency and the 0–1 probability scale, and the fact that the probabilities of an exhaustive set of mutually exclusive outcomes add to 1. From there it builds into listing outcomes systematically with tables, grids, Venn diagrams and tree diagrams, constructing possibility spaces for single and combined experiments, and calculating probabilities for combined events — independent, where the first result does not change the second, and dependent, where it does. Higher adds conditional probability, read from expected frequencies in a two-way table, from a tree diagram, or from a Venn diagram. Almost all of this is natural non-calculator material: the arithmetic is fraction arithmetic, and a tree diagram with the second-stage denominators unchanged is the classic way to lose the marks.
📚 What you learn here
- Recording and analysing outcomes with tables and frequency trees
- Randomness, fairness and equally likely events; expected outcomes over many trials
- Relative frequency, the 0–1 probability scale, and how larger samples approach the theoretical probability
- Probabilities of an exhaustive set of mutually exclusive outcomes summing to 1
- Listing sets and combinations systematically with tables, grids, Venn diagrams and tree diagrams
- Possibility spaces for single and combined experiments with equally likely outcomes
- Independent and dependent combined events, and the assumptions each calculation makes
- Conditional probability from two-way tables, tree diagrams and Venn diagrams (Higher)
- The product rule for counting (Higher)
Frequently asked questions
What is the difference between independent and dependent events?
Events are independent when the first outcome leaves the second unchanged — rolling a die twice, or drawing a counter and replacing it. They are dependent when it does not: drawing two counters without replacement changes both the numerator and the denominator for the second draw.
Do I multiply or add probabilities?
Multiply along the branches of a tree diagram, for one outcome and then another. Add between branches, for one outcome or another, when the outcomes cannot both happen. A probability greater than 1 means you have added when you should have multiplied.
Is conditional probability on Foundation?
No. Foundation covers independent and dependent combined events with tree diagrams; conditional probability as a topic in its own right is Higher only.
What goes in the overlap of a Venn diagram?
The members that are in both sets. Fill the intersection first and then subtract it from each set total, otherwise those members get counted twice and the whole diagram becomes inconsistent with the given totals.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications