Probability - CCEA GCSE Statistics guide to theoretical and experimental probability, tree diagrams, Venn diagrams and expected frequency

Probability is the third knowledge domain of CCEA GCSE Statistics. This guide covers calculating probabilities, the addition and multiplication laws, tree and Venn diagrams, and expected frequency.

The basics

Probability runs from 0 (impossible) to 1 (certain), and the probabilities of all outcomes of an event sum to 1, so the probability of not A is one minus the probability of A. Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes. Experimental probability, or relative frequency, is the number of successes divided by the number of trials, and it becomes a better estimate of the true probability as the number of trials grows. A sample space lists all possible outcomes, such as a two-way table for two dice, making favourable outcomes easy to count.

The addition and multiplication laws

Combined events use two laws. Mutually exclusive events cannot happen together, so for or you add: the probability of A or B is P(A) plus P(B). Independent events do not affect each other, so for and you multiply: the probability of A and B is P(A) times P(B). A simple guide is that or tends to mean add and and tends to mean multiply. Independence matters because once an item is taken without replacement, the next event is no longer independent and its probabilities change.

Tree diagrams

A tree diagram shows two or more events in stages, with probabilities on the branches. You multiply along the branches for a single path and add between paths for the separate ways an outcome can occur. The key skill in without-replacement problems is reducing the second set of probabilities, because removing the first item leaves one fewer, so both the numerator and the denominator usually fall. Listing every path and checking the probabilities on each set of branches sum to 1 keeps the work accurate.

Venn diagrams, expected frequency and risk

A Venn diagram shows how events overlap, with the intersection for and and the union for or, and is ideal for both, either or neither problems. You read a probability by counting the relevant region over the total. Expected frequency predicts how often an event should occur, found by multiplying the probability by the number of trials, and it connects probability to risk, the chance of an outcome interpreted in a real context such as insurance, medicine or quality control.

How CCEA examines probability

CCEA rewards calculating theoretical and experimental probability, using relative frequency and recognising that more trials give a better estimate, applying the addition and multiplication laws correctly, completing and using tree and Venn diagrams including without replacement, and finding expected frequency. Use the dot point for specification-level detail and worked CCEA-style questions, then test yourself with the quiz.