A fair die is rolled. Find P(even number). [1 mark]

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How do we calculate the probability of single and combined events?

Probability of an event, mutually exclusive and independent events, the addition and multiplication rules, Venn diagrams and tree diagrams, and the probability of complementary events.

A CCEA A-Level Mathematics answer on the probability of an event, mutually exclusive and independent events, the addition and multiplication rules, complementary events, and using Venn diagrams and tree diagrams to calculate combined probabilities.

Generated by Claude Opus 4.812 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
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Examples in context
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What this dot point is asking

CCEA wants you to find the probability of an event, recognise mutually exclusive and independent events, apply the addition and multiplication rules, use complementary events, and calculate combined probabilities with Venn diagrams and tree diagrams. Probability is the bridge from descriptive statistics to the binomial distribution in the next dot point.

The answer

Probability of an event

The addition rule

The multiplication rule and independence

Venn and tree diagrams

A Venn diagram displays the overlap of events as regions, making the addition rule visual; fill the intersection first, then the "only" regions, and lastly the outside region for outcomes in neither event. A tree diagram maps a sequence of stages, with probabilities on the branches: multiply along branches for a sequence and add between the relevant end-paths for an "or" outcome. The probabilities on the branches leaving any single point always sum to one, which is a useful check, and for drawing without replacement the second set of branch probabilities differs from the first because the total has shrunk.

Worked example: at least one success

Use the complement for "at least one"

A spinner lands on red with probability $0.3$ on each spin. It is spun three times independently. Find the probability of at least one red.

step Identify the complement

"At least one red" is the complement of "no reds at all", which is easier to compute directly.

step Probability of no reds

Each spin avoids red with probability $1 - 0.3 = 0.7$ , and the spins are independent:

P(\text{no reds}) = 0.7 \times 0.7 \times 0.7 = 0.343.

step Subtract from one

P(\text{at least one red}) = 1 - 0.343 = 0.657.

Using the complement avoids adding the probabilities of exactly one, two and three reds separately.

Examples in context

Example 1. Reliability of a system. A machine works only if both of two independent components work, each reliable with probability $0.95$ . The system reliability is $0.95 \times 0.95 = 0.9025$ , the multiplication rule for independent events, central to engineering design.

Example 2. Medical screening. A Venn diagram of patients testing positive for two conditions makes the overlap explicit, so the proportion with at least one condition uses the addition rule with the overlap subtracted. Visualising the overlap prevents double counting.

Try this

Q1. A fair die is rolled. Find $P(\text{even number})$ . [1 mark]

Cue. $\frac{3}{6} = \frac{1}{2}$ .

Q2. Events $A$ and $B$ are mutually exclusive with $P(A) = 0.3$ and $P(B) = 0.45$ . Find $P(A \cup B)$ . [2 marks]

Cue. $0.3 + 0.45 = 0.75$ .

Q3. A coin is tossed twice. Find the probability of two heads. [2 marks]

Cue. $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ .

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA 20196 marksIn a group of 40 students, 22 study French, 18 study Spanish and 8 study both. Using a Venn diagram, find the probability that a randomly chosen student studies (a) French or Spanish, and (b) neither.

Show worked answer →

Place $8$ in the intersection. French only is $22 - 8 = 14$ ; Spanish only is $18 - 8 = 10$ .

Studying at least one language: $14 + 8 + 10 = 32$ , so $P(\text{French or Spanish}) = \frac{32}{40} = \frac{4}{5} = 0.8$ .

Studying neither: $40 - 32 = 8$ , so $P(\text{neither}) = \frac{8}{40} = \frac{1}{5} = 0.2$ .

Markers reward the intersection, the "only" regions, the probability of at least one, and the complementary probability of neither.

CCEA 20215 marksA bag contains 5 red and 3 blue counters. Two are drawn without replacement. Using a tree diagram, find the probability that both are the same colour.

Show worked answer →

First draw: $P(\text{red}) = \frac{5}{8}$ , $P(\text{blue}) = \frac{3}{8}$ .

Both red: $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56}$ .

Both blue: $\frac{3}{8} \times \frac{2}{7} = \frac{6}{56}$ .

Same colour: $\frac{20}{56} + \frac{6}{56} = \frac{26}{56} = \frac{13}{28}$ .

Markers reward the without-replacement second probabilities, the two matching branches, and adding them for "same colour".

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Sources & how we know this

CCEA GCE Mathematics specification — CCEA (2018)