Probability at GCSE: Everything you need to know

Probability comes up in every GCSE Maths exam. The good news is that most probability questions follow a small number of predictable patterns, and once you learn the methods, the marks are very achievable.

This guide covers everything on the GCSE probability specification: Basic probability, sample spaces, tree diagrams, Venn diagrams, frequency trees, relative frequency, expected outcomes, and conditional probability for higher tier. Each section includes a worked example so you can see the method in action.

Probability measures how likely something is to happen. It is always a value between 0 and 1, where 0 means impossible and 1 means certain. You can also express it as a fraction, decimal, or percentage.

The basic formula is: Probability = number of favourable outcomes / total number of possible outcomes.

For example, a bag contains 3 red balls, 5 blue balls, and 2 green balls. The probability of picking a blue ball at random is 5/10 = 1/2. The total number of outcomes is the total number of balls (10), and the favourable outcomes are the blue ones (5).

Two important rules to remember. The probabilities of all possible outcomes always add up to 1. And the probability of something not happening equals 1 minus the probability of it happening. So if P(blue) = 1/2, then P(not blue) = 1 – 1/2 = 1/2.

A sample space is a complete list of all possible outcomes for an experiment. When two events happen together – like rolling two dice or flipping two coins – a sample space diagram helps you see every possible combination.

Worked example: Two fair dice are rolled and their scores are added together. What is the probability of getting a total of 7?

Draw a grid with die 1 along the top (1 to 6) and die 2 down the side (1 to 6). Fill in each cell with the sum. There are 36 cells in total. Count the cells that show 7: You get (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – that is 6 combinations. So P(total = 7) = 6/36 = 1/6.

Sample space diagrams are useful because they guarantee you do not miss any outcomes. For combined events at foundation tier, this is often the expected method.

Tree diagrams are the go-to method when you have two or more events happening one after the other. Each branch represents a possible outcome, and the probabilities are written along the branches.

The key rules for tree diagrams are straightforward. Multiply along the branches to find the probability of a specific combination. Add between branches to find the probability of one outcome or another. The branches at each stage must add up to 1.

Worked example: A bag contains 4 red counters and 6 blue counters. A counter is taken at random, replaced, and then a second counter is taken. What is the probability of getting two red counters?

First pick: P(red) = 4/10, P(blue) = 6/10. Because the counter is replaced, the second pick has the same probabilities. P(red then red) = 4/10 x 4/10 = 16/100 = 4/25.

If the question says "without replacement", the probabilities change for the second pick. With the same bag, if the first counter is red and not replaced, there are now 3 red and 6 blue left (9 total). So P(red then red without replacement) = 4/10 x 3/9 = 12/90 = 2/15.

Venn diagrams show how two (or more) groups overlap. In probability, each circle represents an event, and the numbers inside represent how many outcomes fall into each category.

Worked example: In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. What is the probability that a randomly chosen student studies French but not Spanish?

Start by filling in the overlap: 5 students study both. French only = 18 – 5 = 13. Spanish only = 12 – 5 = 7. Neither = 30 – 13 – 5 – 7 = 5.

P(French but not Spanish) = 13/30.

Venn diagrams also let you quickly read off other probabilities. P(French or Spanish or both) = (13 + 5 + 7)/30 = 25/30 = 5/6. P(neither) = 5/30 = 1/6. Being comfortable reading probabilities from a completed Venn diagram is a reliable source of marks.

Frequency trees look similar to tree diagrams but use actual counts instead of probabilities. They split a group into categories step by step, and you fill in the numbers at each branch.

Worked example: 200 people were surveyed about whether they exercise regularly. 120 are female. Of the females, 78 exercise regularly. Of the males, 52 exercise regularly.

The tree starts with 200 at the top, splitting into 120 female and 80 male. The female branch splits into 78 (exercise) and 42 (do not exercise). The male branch splits into 52 (exercise) and 28 (do not exercise).

From the completed tree, you can answer questions like: What proportion of people exercise regularly? That is (78 + 52)/200 = 130/200 = 13/20.

Frequency trees are particularly useful when a question gives you a mixture of totals and sub-totals, because you can work out missing values by subtraction.

Relative frequency is an estimate of probability based on experimental data rather than equally likely outcomes. You calculate it the same way – favourable outcomes divided by total trials – but it comes from actual results.

For example, a spinner is spun 200 times and lands on red 56 times. The relative frequency of red is 56/200 = 0.28. This is an estimate, not an exact probability, and it becomes more reliable as the number of trials increases.

Expected outcomes use probability to predict how many times something should happen. Multiply the probability by the number of trials.

Worked example: A biased coin has a probability of 0.6 of landing on heads. If the coin is flipped 150 times, how many heads would you expect? Expected heads = 0.6 x 150 = 90.

A common exam question gives you relative frequency data and then asks you to calculate expected outcomes for a different number of trials. Use the relative frequency as your probability estimate and multiply by the new number of trials.

Conditional probability is the probability of one event happening given that another event has already happened. It appears only at higher tier and is often combined with tree diagrams or Venn diagrams.

The notation P(A | B) means "the probability of A given that B has happened". The formula is P(A | B) = P(A and B) / P(B).

Worked example using a Venn diagram: In the class of 30 students from earlier, 18 study French, 12 study Spanish, and 5 study both. Given that a student studies Spanish, what is the probability that they also study French?

P(French | Spanish) = P(French and Spanish) / P(Spanish) = (5/30) / (12/30) = 5/12.

You can also think of it more directly: There are 12 Spanish students, and 5 of them also study French, so 5/12.

Conditional probability also appears in tree diagram questions without replacement. When you draw the second set of branches after removing an item, you are already applying conditional probability – the probabilities depend on what happened first.