GCSE Maths · Probability · Higher · Summary

GCSE Maths Probability (Higher): summary and key words

Part of the GCSE Maths probability chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Probability of an event	favourable outcomes ÷ total outcomes, for equally likely outcomes. Always between 0 and 1.
Probability of not A	P(not A) = 1 − P(A). All the probabilities of the outcomes add up to 1.
Relative frequency	frequency ÷ number of trials. It estimates the probability. More trials give a better estimate.
Expected number	probability × number of trials
Fair or biased?	Compare relative frequencies with the theoretical probability, using plenty of trials. Fair does not mean exactly equal.
Listing outcomes	List systematically or use a sample-space table. Probability = matching cells ÷ all cells (two dice: 36).
Counting outcomes	Product rule: multiply the number of choices at each stage.
Independent events	P(A and B) = P(A) × P(B). “And” means multiply.
Mutually exclusive events	P(A or B) = P(A) + P(B). “Or” means add.
At least one	P(at least one) = 1 − P(none)
Tree diagrams	Multiply along the branches, add the routes. Without replacement the numbers on the second branches change.
Sets and Venn diagrams	∩ both, ∪ either, A′ not in A, ξ everything. Fill in from the middle outwards. P = n(event) ÷ n(ξ).
Two-way tables	Rows and columns add up to their totals. Probability = number ÷ grand total.
Frequency trees	Numbers, not probabilities. Take a fraction of the number in the box before it.
Higher: P(A or B)	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Higher: conditional probability	P(A \| B) = P(A and B) ÷ P(B) = n(A ∩ B) ÷ n(B). “Given” shrinks the sample space.
Higher: probability and algebra	Write each probability with the unknown, form an equation (probabilities add to 1, or a route probability equals a given value) and solve.

Can you…?

I can use the words impossible, unlikely, even chance, likely and certain, and the probability scale from 0 to 1.
I can work out the probability of a single event when outcomes are equally likely.
I can use P(not A) = 1 − P(A) and find a missing probability.
I can work out a relative frequency and use it to estimate a probability.
I can find an expected number of outcomes.
I can say whether an experiment suggests a die or coin is fair or biased, and explain why more trials are better.
I can list outcomes systematically and use a sample-space table.
I can use the product rule to count outcomes.
I can use the AND rule for independent events and the OR rule for mutually exclusive events.
I can draw and use tree diagrams with and without replacement.
I can use set notation and Venn diagrams to count and to find probabilities.
I can complete and use two-way tables and frequency trees.
Higher: I can find conditional probabilities from tables, Venn diagrams and tree diagrams.
Higher: I can form and solve equations from probability information.

Key terms

Probability: A number from 0 to 1 that measures how likely an event is.
Outcome: One possible result of an experiment.
Event: An outcome or group of outcomes we are interested in.
Equally likely: Outcomes that each have the same chance of happening.
Random: Not predictable or rigged: each outcome has its fair chance.
Biased: Not fair: some outcomes are more likely than they should be.
Theoretical probability: Probability worked out from equally likely outcomes: favourable ÷ total.
Trial: One go of an experiment.
Frequency: The number of times an event happens.
Relative frequency: Frequency divided by the number of trials: an estimate of probability.
Expected number: Probability multiplied by the number of trials.
Complement: The event “A does not happen”, written not A or A′.
Exhaustive: Covering every possible outcome, so the probabilities add up to 1.
Mutually exclusive: Events that cannot happen at the same time.
Independent: Events where one has no effect on the probability of the other.
Dependent: Events where the result of one changes the probability of the other.
Sample space: The list or table of all possible outcomes.
Product rule for counting: Number of outcomes = (choices at stage 1) × (choices at stage 2) × ...
Tree diagram: A diagram of branches showing the outcomes of events in order, with their probabilities.
With replacement: An item is put back after it is picked, so probabilities stay the same.
Without replacement: An item is not put back, so later probabilities change.
Set: A collection of items called elements, written in curly brackets.
Universal set (ξ): The set of everything being considered.
Intersection (A ∩ B): The elements in both A and B.
Union (A ∪ B): The elements in A or B or both.
Venn diagram: A diagram that uses overlapping circles inside a rectangle to show sets.
Two-way table: A table showing two categories at once, with row and column totals.
Frequency tree: A tree showing numbers of people or items instead of probabilities.
Conditional probability: The probability of an event given that another event has happened: P(A | B).