Eduqas GCSE Mathematics Probability: a complete overview of the probability scale, tree diagrams, Venn diagrams and expected outcomes

What the Probability content demands

Probability mixes careful counting with clear reasoning. Eduqas tests both the calculation and the interpretation (what a probability means, why more trials improve an estimate, what an expected value predicts), so a fluent grasp of the underlying ideas matters as much as the arithmetic. The content runs from the probability scale and single events through tree and Venn diagrams to relative frequency and expected outcomes, with conditional probability (without replacement, and the conditional Venn reading) the main Higher-tier emphasis. Fractions, decimals and percentages are used throughout, so number fluency underpins it.

This guide walks through the four areas of the content and ties together the matching dot-point pages, each with its own practice questions.

Probability basics

Probability runs from $0$ (impossible) to $1$ (certain). For equally likely outcomes, P(event) $= \dfrac{\text{favourable}}{\text{total}}$. All outcomes' probabilities sum to $1$, so P(not A) $= 1 -$ P(A), which makes the complement a powerful shortcut for "at least one" questions. A sample space diagram lists every outcome of two combined events (such as two dice in a $6 \times 6$ grid), making favourable outcomes easy to count.

Tree diagrams

A tree diagram shows combined events as branches. Multiply along a path (AND) and add between paths (OR). For independent events (with replacement), the second branch probabilities match the first. At Higher tier, without replacement makes the second probabilities conditional: reduce the total, and the matching numerator, after the first item is removed. The "at least one" question is often quickest via the complement.

Venn diagrams and set notation

A Venn diagram shows overlapping sets: the intersection $A \cap B$ is the overlap, the union $A \cup B$ is the whole of both circles, and the complement $A'$ is everything outside A. Fill the overlap first, then the "only" regions, then "neither". Probabilities are the region count over the total. At Higher tier, conditional probability restricts the sample space to the given condition, changing the denominator.

Relative frequency and expected outcomes

Relative frequency estimates a probability from data: $\dfrac{\text{frequency}}{\text{trials}}$, the only route when an item is biased. More trials give a better estimate because random variation has less effect. Expected outcomes multiply the probability by the number of trials, predicting how many times an event should occur, though the actual count varies around it.

Check your knowledge

A mix of single-event, tree, Venn and expected-outcome questions. Attempt them under timed conditions, then check against the solutions.

1. A bag has 4 red, 5 blue and 1 green counter. Find P(blue). (1 mark)
2. P(rain) is $0.35$. Find P(no rain). (1 mark)
3. Two fair coins are flipped. Find P(two heads). (2 marks)
4. A bag has 3 white and 7 black balls. One is drawn. Find P(white). (1 mark)
5. In a group of 25, 12 like tea, 9 like coffee and 4 like both. How many like neither? (3 marks)
6. A spinner lands on red 30 times in 120 spins. Estimate P(red). (2 marks)
7. With P(six) $= \tfrac{1}{6}$, find the expected number of sixes in 240 rolls. (2 marks)
8. Two balls are drawn without replacement from a bag of 2 red and 3 blue. Find P(both red). (2 marks)