Key Stage 3 maths · Probability

KS3 probability, sample spaces and Venn diagrams: a guide

Probability measures how likely something is on a scale from 0 (impossible) to 1 (certain). In Years 7 to 9 pupils run simple experiments and record the frequency of outcomes, learn that all the probabilities of a situation add to 1, list outcomes systematically in tables and grids, use Venn diagrams to organise sets, and calculate theoretical probabilities for single and combined events.

Probability and Venn diagrams, explained

Probability is described in words (impossible, unlikely, evens, likely, certain) and in numbers on a scale from 0 to 1. A probability can be written as a fraction, a decimal or a percentage, so 1/4, 0.25 and 25% mean the same thing. It cannot be bigger than 1 or smaller than 0. Pupils are expected to use this language precisely and place events on the scale, justifying their choices.

Experiments come first. Pupils toss coins, roll dice, spin spinners or draw counters from a bag, then record the frequency of each outcome and calculate the experimental probability (also called relative frequency), which is the number of times something happened divided by the number of trials. They learn that results vary from one set of trials to another, and that more trials usually bring the results closer to what theory predicts. They also meet the idea of fairness: a fair die has equally likely outcomes, while a biased one does not, and a record of many throws can suggest whether a die or spinner is biased.

When the outcomes are equally likely and cannot happen at the same time (mutually exclusive), theoretical probability is the number of ways an event can happen divided by the total number of outcomes. All the probabilities of the outcomes of an experiment must add up to 1. This gives a useful shortcut: if the probability that it rains is 0.3, the probability that it does not rain is 1 − 0.3 = 0.7. Pupils use it constantly, particularly when the opposite event is easier to count.

Organising possibilities is a skill in itself. For combined events, such as flipping two coins or adding two dice, pupils build a sample space, the list of every possible outcome, using a table, grid or systematic list. Counting from the grid avoids missing combinations. Venn diagrams work for sets: a union (everything in at least one set), an intersection (what is in both) and the region outside both. In a class of 30, if 14 pupils swim, 9 cycle and 5 do both, then 14 + 9 − 5 = 18 do at least one of the two, and 30 − 18 = 12 do neither. Pupils must fill the intersection first, so nobody is counted twice.

Words your child will hear

Outcome: One possible result of an experiment, such as rolling a 4.
Event: One or more outcomes we are interested in, such as rolling an even number.
Sample space: The complete list of all possible outcomes, often shown as a table or grid.
Mutually exclusive: Events that cannot happen at the same time, such as rolling a 2 and rolling a 5 on one throw.
Relative frequency: The fraction of trials in which an outcome happened. It estimates probability from an experiment.
Union and intersection: The items in either set (union) or in both sets (intersection).
Biased: Not fair: some outcomes are more likely than others.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale
Understand that the probabilities of all possible outcomes sum to 1
Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Two fair six-sided dice are rolled and their scores are added. Find the probability that the total is 8, and the probability that it is not 8.

List every outcome in a grid. The first die gives the row and the second die the column, and each cell holds the total. There are 6 × 6 = 36 equally likely outcomes.
```
    1  2  3  4  5  6
1   2  3  4  5  6  7
2   3  4  5  6  7  8
3   4  5  6  7  8  9
4   5  6  7  8  9 10
5   6  7  8  9 10 11
6   7  8  9 10 11 12
```
Count the cells with a total of 8. They are (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2), so there are 5.
Divide the number of successful outcomes by the total.
```
P(8) = 5/36
```
The probabilities of all outcomes add up to 1, so the probability of not getting 8 is 1 minus this.
```
P(not 8) = 1 − 5/36
         = 31/36
```

P(total is 8) = 5/36, and P(total is not 8) = 31/36.

Common mistakes and how to help

Treating all dice totals as equally likely

The totals run from 2 to 12, which is 11 possibilities, so it seems each has a probability of 1/11.

What helps: Build the grid. A total of 7 appears 6 times but 2 appears once. The equally likely things are the 36 pairs of scores, not the totals.

Believing a result is “due”

After five heads in a row, many people feel tails must come next. But the coin has no memory.

What helps: Run a coin experiment and look at what comes after streaks. The probability of heads on the next toss is still 1/2.

Comparing successes with failures, such as giving 1/5 for rolling a six

“One to five” (the 1 way it happens against the 5 ways it does not) is mixed up with “one in six” (1 way out of all 6 outcomes).

What helps: Probability always compares the successful outcomes with all the outcomes. For a six on a die, the answer is 1 out of 6 possibilities, so 1/6.

Counting the overlap twice in a Venn diagram

Adding 14 swimmers and 9 cyclists gives 23 pupils, but 5 appear in both groups, so they have been counted twice.

What helps: Fill in the intersection first, subtract it from each set to find the “only” regions, then add the totals. Check that everything adds up to the number of people altogether.

How to help at home

Test a die. Roll a die 60 times and tally each score. About 10 of each is expected, but you will not get exactly that. Roll 60 more and combine the results. Does it get closer?
Build a dice grid for a board game. Before a game that uses two dice, ask which total will come up most often. Use the grid to find out and test it as you play.
Make a family Venn diagram. Draw two overlapping circles and sort family members or friends by, say, whether they like tea, coffee, both or neither. Count each region and check the total.
Compare forecasts with outcomes. Weather forecasts give the chance of rain as a percentage. Keep a week’s record of the forecast against what happened and discuss what 30% really means.

Leads on to

Key Stage 4: Tree diagrams and combined events
Key Stage 4: Conditional probability

Everything in Key Stage 3 maths

Updated 20 October 2026.