What you'll learn
Combined events involve the probability of two or more things happening, and the ideas of possibility spaces and mutually exclusive and exhaustive events. In this guide you will learn how to list outcomes with possibility spaces, how to use the rules for mutually exclusive and exhaustive events, how to add and multiply probabilities, and how to solve combined-event problems. These ideas are central to GCSE probability.
Key terms and definitions
Combined events — two or more events considered together.
Possibility space — a diagram or table showing all possible outcomes of combined events.
Mutually exclusive events — events that cannot happen at the same time.
Exhaustive events — events that together cover all possible outcomes.
Independent events — events where one does not affect the other.
Core concepts
Possibility spaces
A possibility space (or sample space diagram) lists or maps all possible outcomes of combined events, often in a table. For example, rolling two dice gives a 6 × 6 grid of 36 outcomes. From the possibility space you can count the outcomes that meet a condition and find the probability as favourable outcomes ÷ total outcomes.
Mutually exclusive events
Mutually exclusive events cannot occur at the same time — for example, rolling a 2 and rolling a 5 on a single die. For mutually exclusive events, the probability of one or the other is found by adding: P(A or B) = P(A) + P(B).
Exhaustive events
Exhaustive events together cover all possible outcomes, so their probabilities add up to 1. If a set of events is both mutually exclusive and exhaustive, then P(A) + P(B) + … = 1. This lets you find a missing probability by subtracting from 1.
Independent events and multiplying
For independent events (one does not affect the other), the probability of both happening is found by multiplying: P(A and B) = P(A) × P(B). For example, the probability of getting heads on a coin and a 6 on a die is ½ × ⅙ = 1/12.
"And" versus "or"
A key distinction: use multiply for "and" (both events happening, independent) and add for "or" (mutually exclusive events). Read the question carefully to decide which applies.
Worked examples
Example 1: Possibility space
Two coins are flipped. What is the probability of two heads?
Outcomes: HH, HT, TH, TT (4 total). Two heads is one outcome, so P = 1/4.
Example 2: Mutually exclusive
A bag has red, blue and green counters with P(red) = 0.3 and P(blue) = 0.5. What is P(green)?
The events are exhaustive, so they sum to 1: P(green) = 1 − 0.3 − 0.5 = 0.2.
Example 3: Independent events
A coin is flipped and a die is rolled. Find P(tails and a 4).
Independent, so multiply: ½ × ⅙ = 1/12.
Common mistakes and how to avoid them
Adding when you should multiply. Multiply for "and" (independent events); add for "or" (mutually exclusive).
Forgetting probabilities sum to 1. Use this for exhaustive events to find a missing value.
Miscounting the possibility space. List or tabulate all outcomes carefully.
Assuming events are mutually exclusive when they aren't. The simple addition rule only applies if they cannot overlap.
Treating dependent events as independent. If one event changes the next, the probabilities change.
Exam technique for Combined Events
Draw a possibility space to count outcomes for combined events.
Identify mutually exclusive events and add their probabilities.
Use the sum-to-1 rule for exhaustive events.
Multiply for independent "and" events, add for "or" events.
Read the question to decide between adding and multiplying.
Quick revision summary
Combined events consider two or more events together. A possibility space lists all outcomes (e.g. a 6 × 6 grid for two dice), and probability = favourable ÷ total. Mutually exclusive events cannot happen together, so P(A or B) = P(A) + P(B) (add). Exhaustive events cover every outcome, so their probabilities sum to 1 — handy for finding a missing probability by subtracting from 1. Independent events don't affect each other, so P(A and B) = P(A) × P(B) (multiply). The crucial rule: multiply for "and" (independent) and add for "or" (mutually exclusive). Common errors are mixing up add and multiply, forgetting the sum-to-1 rule, miscounting outcomes, wrongly assuming exclusivity, and treating dependent events as independent. Draw a possibility space when in doubt, identify whether events are exclusive or independent, apply the right operation, and use the fact that exhaustive probabilities total 1.