Basic Probability and Sample Space — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers calculating the probability of an event, listing the possible outcomes, and using sample space diagrams for two events happening together. By the end of this guide you should be able to place a probability on the 0-to-1 scale, calculate one for equally likely outcomes, and use the rule that probabilities sum to 1.

You should also be able to list outcomes systematically so that none are missed, read probabilities off a sample space diagram, find a missing probability from a table, and estimate how often an event will occur in a given number of trials.

The organising idea is that a probability is a fraction of a complete list. The favourable outcomes go on top and all the possible outcomes go underneath — so the real work is almost never the fraction itself, it is making sure the list underneath is complete and that every item on it is equally likely. Nearly every lost mark in this topic comes from an incomplete list, which is why systematic listing and sample space diagrams matter so much.

Key terms and definitions

Trial — one go at something uncertain, such as a single roll of a die.

Outcome — one possible result of a trial.

Event — the result you are interested in, which may cover several outcomes.

Sample space — the complete list of all possible outcomes.

Equally likely — outcomes with the same chance, as on a fair die or coin.

Mutually exclusive — events that cannot both happen at once.

Exhaustive — a set of outcomes covering every possibility, so their probabilities total 1.

Relative frequency — a probability estimated from results actually observed, rather than from theory.

Core concepts

The probability scale

Every probability lies between 0 (impossible) and 1 (certain), and may be written as a fraction, a decimal or a percentage. An even chance is 1/2, 0.5 or 50%.

A probability can never be negative or greater than 1. An answer such as 1.4 or −0.2 is a signal that something has gone wrong, and spotting that takes a second.

Words map onto the scale loosely: "unlikely" sits below a half, "likely" above it.

Calculating a probability

When outcomes are equally likely:

P(event) = number of favourable outcomes ÷ total number of outcomes

Rolling a 3 on a fair die gives 1/6, since one face out of six is favourable. Rolling an even number gives 3/6, which simplifies to 1/2, since three faces qualify.

The words fair and unbiased are what tell you the outcomes are equally likely. Without them the formula does not apply — a bent coin or a weighted spinner needs its probabilities given or estimated from data.

Probabilities that sum to 1

Something must happen, so the probabilities of all possible outcomes add to 1.

Two consequences follow, and both are heavily used.

P(not A) = 1 − P(A). If the probability of rain is 0.3, the probability of no rain is 0.7. This is much quicker than adding up all the other cases, and it is the key to "at least one" questions.

A missing probability can be found by subtraction. If a spinner's four colours have probabilities 0.2, 0.3 and 0.1 with one unknown, the missing one is 1 − 0.6 = 0.4.

Mutually exclusive events

Two events are mutually exclusive when they cannot both happen in the same trial — rolling a 2 and rolling a 5, for instance.

For such events the probabilities simply add: the chance of rolling a 2 or a 5 is 1/6 + 1/6 = 2/6.

Adding only works when the events are mutually exclusive. Drawing a red card and drawing a king are not, because the red kings belong to both, and adding would count them twice.

Listing outcomes systematically

Listing at random almost guarantees a missed outcome. Work in a fixed order instead.

For two coins, run through the first coin's results in turn and pair each with both of the second's: HH, HT, TH, TT. Four outcomes, and TH and HT are genuinely different, which is the usual omission.

For a three-digit number made from the digits 1, 2 and 3 without repeats, fix the first digit and vary the rest: 123, 132, then 213, 231, then 312, 321. Six outcomes, found in an order that makes a gap visible.

Sample space diagrams

For two events together, a grid is faster and safer than a list. Put one event along the top and the other down the side, then fill each cell with the combined result.

Two dice give a 6 by 6 grid with 36 cells. Filling it with the totals shows immediately that 7 appears six times, so the probability of a total of 7 is 6/36, which is 1/6.

Two things are worth noting. The number of cells is the two totals multiplied, so two dice give 36 rather than 12. And the totals are not equally likely even though the cells are — there is only one way to make 2 but six ways to make 7. Always count cells, never possible totals.

Expected frequency

To estimate how many times an event will happen in a number of trials, multiply the probability by the number of trials.

Rolling a fair die 60 times gives an expected 1/6 × 60 = 10 sixes.

This is an expectation, not a guarantee. Getting 8 or 13 sixes in 60 rolls is perfectly ordinary, and questions often ask you to say so.

Theoretical and experimental probability

A theoretical probability comes from the structure of the situation — six equally likely faces on a die.

A relative frequency comes from results actually recorded: the number of times the event happened divided by the number of trials.

The more trials, the closer the relative frequency usually gets to the theoretical value. A large difference after many trials is evidence that the object is biased.

Worked examples

Example 1: A missing probability

A spinner has four colours. P(red) = 0.2, P(blue) = 0.35 and P(green) = 0.15. Find the probability of yellow, then the probability of not landing on red.

All four probabilities must total 1. The three given add to 0.2 + 0.35 + 0.15 = 0.7.

So P(yellow) = 1 − 0.7 = 0.3.

For the second part, use the rule for the complement: P(not red) = 1 − 0.2 = 0.8.

Adding the other three probabilities would give the same answer, but the subtraction is one step instead of three.

Example 2: A sample space diagram

Two fair dice are rolled and their scores added. Find the probability that the total is 9.

The grid has 6 rows and 6 columns, so there are 36 equally likely cells.

Totals of 9 come from 3 and 6, 4 and 5, 5 and 4, and 6 and 3 — four cells, because the two dice are distinguishable.

So P(total is 9) = 4/36 = 1/9.

Counting only "3 and 6" and "4 and 5" as two possibilities would halve the answer, which is the standard error with two dice.

Example 3: Expected frequency

A bag holds 3 red, 5 blue and 2 green counters. One is drawn at random and replaced, 50 times. How many greens would you expect?

There are 10 counters altogether, so P(green) = 2/10 = 1/5.

Expected number = 1/5 × 50 = 10.

Because the counter is replaced each time, the probability stays the same throughout, which is what makes the simple multiplication valid.

Common mistakes and how to avoid them

An incomplete list of outcomes. List in a fixed order, or draw a grid. HT and TH are different outcomes.

Using the wrong total. The denominator is all possible outcomes, not just the ones mentioned in the question.

Treating totals as equally likely. On two dice the 36 cells are equally likely; the totals are not.

Adding probabilities of events that overlap. Adding works only for mutually exclusive events.

Giving a probability above 1 or below 0. Check every answer lies on the scale.

Confusing expected frequency with a guarantee. Ten sixes in 60 rolls is what you expect on average, not what must happen.

Applying the equally-likely formula to a biased object. Look for the words fair or unbiased before using it.

Exam technique for "Basic Probability"

Write the total number of outcomes down first. It fixes the denominator and is often worth a mark by itself.

Draw the sample space grid whenever two events are combined. It is quicker than listing and far harder to get wrong.

Use 1 − P(A) whenever a question asks for "not" or "at least one". It replaces several calculations with one.

Give the answer in the form the question asks for, and simplify a fraction unless told otherwise.

Check the answer lies between 0 and 1 before moving on.

When asked to comment on whether something is fair, compare the relative frequency with the theoretical probability and mention the number of trials — a difference over few trials proves nothing.

Quick revision summary

A probability is a fraction of a complete list: favourable outcomes over all possible outcomes, and the list must be complete with every item equally likely.

Probabilities run from 0 to 1 and can never fall outside that range.

P(event) = favourable ÷ total applies when outcomes are equally likely, which the words fair or unbiased signal.

All probabilities sum to 1, so P(not A) = 1 − P(A), and a missing probability is found by subtracting the rest from 1.

Mutually exclusive events cannot both happen, and only then may their probabilities be added.

List outcomes systematically, and use a sample space diagram for two events. Two dice give 36 equally likely cells — but the totals are not equally likely, so count cells.

Expected frequency = probability × number of trials, and it is an average, not a guarantee.

Relative frequency estimates a probability from observed results, and approaches the theoretical value as the number of trials grows.