Tree diagrams, Venn diagrams, conditional probability and more — all the probability skills you need for GCSE Maths.
Probability is one of the topics where students lose marks unnecessarily. The concepts aren't especially difficult — but the question types are very varied, and students who haven't seen all of them before tend to get thrown off in the exam. This guide covers every probability question type you'll encounter at GCSE, with full worked examples.
Before anything else, these core rules underpin all of probability at GCSE:
The complement rule sounds obvious but it's used in a surprising number of questions — particularly ones that ask for the probability that something does NOT happen, or "at least one" of something happens (where it's easier to find the probability of zero happening and subtract from 1).
For simple experiments with two components — rolling two dice, spinning two spinners, picking two items — you should be able to construct a possibility space diagram. This is a grid showing all possible outcomes.
Example: Two fair dice are rolled. Find the probability that the sum is greater than 9.
A 6×6 grid gives 36 total outcomes. Outcomes where the sum > 9 are: (4,6), (5,5), (5,6), (6,4), (6,5), (6,6) — that's 6 outcomes. So P(sum > 9) = 6/36 = 1/6.
The power of the possibility space is that you don't have to rely on memory or mental arithmetic — you can count directly. Draw the grid even if it seems slow. On a 3-mark question, accuracy matters more than speed.
Tree diagrams are used when events happen in sequence, and you want to find the probability of a combination of outcomes. They work for both independent events (where one outcome doesn't affect the next) and dependent events (where it does).
Multiply along branches (AND). Add between branches (OR). This rule covers virtually every tree diagram question at GCSE.
A bag contains 3 red and 2 blue balls. A ball is picked at random and replaced. A second ball is then picked. Find the probability of picking one red and one blue ball (in either order).
Because the ball is replaced, the probabilities don't change on the second pick. P(red) = 3/5, P(blue) = 2/5 each time.
P(red then blue) = 3/5 × 2/5 = 6/25
P(blue then red) = 2/5 × 3/5 = 6/25
P(one of each) = 6/25 + 6/25 = 12/25
Same bag: 3 red, 2 blue. This time no replacement. Find P(both balls are the same colour).
First pick is red (P = 3/5). Now 4 balls remain: 2 red, 2 blue. Second pick is also red: P = 2/4 = 1/2.
P(both red) = 3/5 × 1/2 = 3/10
First pick is blue (P = 2/5). Now 4 balls remain: 3 red, 1 blue. Second pick is also blue: P = 1/4.
P(both blue) = 2/5 × 1/4 = 2/20 = 1/10
P(same colour) = 3/10 + 1/10 = 4/10 = 2/5
⚠️ Without replacement: the denominator drops by 1 for the second pick, AND the numerator changes depending on what was picked first. Draw a full tree diagram and label every branch to keep track.
Venn diagrams are tested heavily, especially at Higher tier. They show two or more sets and their overlaps. The key is filling in the numbers correctly before answering any probability questions.
In a class of 30 students: 18 study French, 14 study Spanish, 6 study both. Draw a Venn diagram and find P(student studies French only).
French only = 18 − 6 = 12. Spanish only = 14 − 6 = 8. Both = 6. Neither = 30 − 12 − 6 − 8 = 4.
Check: 12 + 8 + 6 + 4 = 30 ✓
P(French only) = 12/30 = 2/5
Before answering any probability questions from a Venn diagram, fill in every region's value. If your numbers don't add up to the total given, something is wrong. Catching this before you answer the probability parts saves you from carrying errors through multiple questions.
Conditional probability is the probability of an event occurring given that another event has already occurred. It appears at Higher tier and is one of the topics students find most confusing. But there's a formula that makes it straightforward:
In practice, conditional probability questions often use Venn diagrams or frequency tables rather than the formula directly — you just need to restrict your sample space to the condition that's been given.
Using the French/Spanish Venn diagram from above. Given that a student studies French, what is the probability they also study Spanish?
We're told the student studies French, so our sample space is now just the 18 French students. Of those 18, 6 also study Spanish.
P(Spanish | French) = 6/18 = 1/3
Notice: we didn't use the formula directly. We just restricted our view to the French group and counted within that group. This approach works for almost every conditional probability question at GCSE.
Frequency trees show frequencies rather than probabilities, but you can extract probabilities from them. They're similar to tree diagrams but show actual counts. Two-way tables organise data into rows and columns covering two categorical variables.
For both: find the total for the relevant row or column first before working out a probability. The most common mistake is using the wrong total — using the overall total when you should be using a row or column total, or vice versa.
The AQA probability specification, including the full list of assessed skills, is available on the AQA GCSE Mathematics specification page. Edexcel's equivalent breakdown is on the Edexcel GCSE Mathematics page.
PaperPlus generates unlimited GCSE probability questions across all types — tree diagrams, Venn diagrams, conditional probability — with full mark schemes for AQA, Edexcel and OCR.
Start Practising Free →Adding when you should multiply. If two events both need to happen (AND), multiply the probabilities. If either one happening counts (OR), add. Getting this backwards on a tree diagram is the single most costly probability error at GCSE.
Not checking branches sum to 1. After drawing a tree diagram, check every set of branches from a single point adds to 1. If they don't, you've made an error and every probability calculated from that point onwards will be wrong.
Forgetting to simplify fractions. Probability answers should be given as simplified fractions, decimals, or percentages — whichever the question asks for. Leaving 6/36 instead of simplifying to 1/6 often loses the final mark.
Using the wrong denominator in conditional probability. Always restrict your sample space to the given condition. The denominator is the size of the restricted group, not the full total.