OCR GCSE Mathematics Probability: a complete overview of the probability scale, tree diagrams, Venn diagrams and relative frequency

What the Probability content demands

Probability applies number and reasoning skills to uncertainty, and it rewards careful counting, exact fractions and clear logic. The content runs from the probability scale and single events through tree and Venn diagrams to relative frequency and expected outcomes. Because OCR puts the non-calculator paper in the middle of the three, the fraction work in probability must be fluent by hand, and the reasoning ("exactly one", "given that") carries AO2 marks.

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

The probability scale and single events

Probability runs from $0$ (impossible) to $1$ (certain) and can be a fraction, decimal or percentage. For equally likely outcomes, the probability of an event is the number of favourable outcomes over the total number of outcomes. All the probabilities of an event's outcomes sum to $1$, so $P(\text{not } A) = 1 - P(A)$. A sample space diagram lists every combined outcome of two events, making favourable cases easy to count.

Tree diagrams

A tree diagram organises a sequence of events, with branch probabilities summing to $1$ at each stage. Multiply along a path (the AND rule) for the probability of a sequence, and add across paths (the OR rule) to combine outcomes. For independent events the branch probabilities stay the same; without replacement they change, because the total drops by one after the first pick and the favourable count may drop too.

Venn diagrams and set notation

A Venn diagram sorts items into overlapping circles. The intersection $A \cap B$ is the overlap, the union $A \cup B$ is everything in either set, and the complement $A'$ is everything outside $A$. Fill the overlap first, then the "only" regions, then the outside. Probabilities are counts over the total, and conditional probability restricts the total to a given group.

Relative frequency and expected outcomes

Relative frequency (experimental probability) is the number of successes over the number of trials, and it estimates the true probability more reliably the more trials there are. If the relative frequency settles away from the theoretical value, the object is biased. The expected number of outcomes is the probability multiplied by the number of trials.

Check your knowledge

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

1. A bag has $4$ red and $6$ yellow counters. Find $P(\text{red})$. (1 mark)
2. $P(\text{rain}) = 0.35$. Find $P(\text{no rain})$. (1 mark)
3. Two fair coins are flipped. Find the probability of two heads. (2 marks)
4. A bag has $3$ green and $5$ blue. Two are taken without replacement. Find $P(\text{both green})$. (3 marks)
5. Sets $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. List $A \cap B$. (1 mark)
6. In a group of $20$, $12$ like tea, $9$ like coffee, $5$ like both. How many like neither? (3 marks)
7. A biased die shows a five $30$ times in $120$ rolls. Estimate $P(\text{five})$. (2 marks)
8. Using that estimate, find the expected number of fives in $400$ rolls. (2 marks)