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

What the Probability content demands

Probability rewards correct method and clear interpretation. WJEC examines a small set of reliable techniques (the probability scale, sample spaces, tree diagrams, Venn diagrams and expected frequencies), and the marks turn on applying the right rule (multiply for "and", add for "or") and on handling combined events without missing a case. Because Unit 1 is non-calculator, fraction work matters, while Unit 2 brings decimals and larger experiments. This guide walks through the content and links to the matching dot-point pages, each with its own practice questions.

Probability basics

A probability is a number from $0$ to $1$. For equally likely outcomes it is favourable outcomes over total outcomes. All outcomes sum to $1$, so $P(\text{not A}) = 1 - P(A)$. Mutually exclusive events cannot happen together, so $P(A \text{ or } B) = P(A) + P(B)$. Sample space diagrams and two-way tables list all combined outcomes so they can be counted, for example the $36$ outcomes of two dice.

Tree diagrams

A tree diagram shows two or more events as branches. Multiply probabilities along a branch (the "and" of successive events) and add the probabilities of the routes that satisfy the condition (the "or"). For independent events (with replacement) the probabilities stay fixed; without replacement (Higher) they change, because an item has been removed, so reduce both the count and the total for the second draw. "Multiply along, add across" is the whole method.

Venn diagrams and set notation (Higher)

A Venn diagram shows sets as overlapping circles inside a rectangle. The intersection $A \cap B$ is the overlap ("and"), the union $A \cup B$ is everything in either ("or"), and the complement $A'$ is everything not in A. Fill it in by placing the intersection first, then subtracting it from each set total to find the "only" regions, and put the rest outside. Probabilities are read as a region's count over the total.

Relative frequency and expected outcomes

Relative frequency, $\tfrac{\text{successes}}{\text{trials}}$, estimates probability from experiment and is used when outcomes are not equally likely (a biased object). The more trials, the more reliable the estimate. The expected frequency of an event is its probability times the number of trials, $P(\text{event})\times n$, which predicts how often it should occur. "Expected" is a prediction of the average, not a guarantee.

Check your knowledge

A mix of scale, sample space, tree, Venn and expected frequency questions covering the Probability content. Attempt them under timed conditions, then check against the solutions.

1. A bag has $4$ red and $6$ blue counters. Find $P(\text{red})$. (1 mark)
2. If $P(\text{rain}) = 0.3$, find $P(\text{no rain})$. (1 mark)
3. Two fair coins are flipped. Find the probability of two heads. (2 marks)
4. A spinner lands on green with probability $0.2$. Spun $50$ times, how many greens are expected? (2 marks)
5. A biased dice lands on six $30$ times in $120$ rolls. Estimate $P(\text{six})$. (2 marks)
6. In a Venn diagram, $A \cap B$ describes which region? (1 mark)
7. A bag has $3$ green and $5$ red beads. Two are drawn without replacement. Find $P(\text{both green})$. (3 marks)
8. State the rule for combining probabilities along a tree branch. (1 mark)