What you'll learn
Combined events occur when two or more events happen together or in sequence. This topic builds on single-event probability by examining how to calculate probabilities when multiple outcomes interact. You'll master tree diagrams, two-way tables, sample space diagrams, and understand when events are independent or mutually exclusive—skills directly tested in AQA GCSE Statistics examinations.
Key terms and definitions
Independent events — two events where the outcome of one does not affect the probability of the other (e.g., rolling a die twice, spinning two separate spinners)
Dependent events — two events where the outcome of the first affects the probability of the second (e.g., picking two cards without replacement from a deck)
Mutually exclusive events — events that cannot happen at the same time (e.g., rolling a 3 and rolling a 5 on a single die roll)
Tree diagram — a branching diagram showing all possible outcomes of two or more events with probabilities on each branch
Sample space diagram — a table or grid showing all possible outcomes when two events occur
Conditional probability — the probability of an event occurring given that another event has already occurred
Combined probability — the probability that two or more events all occur, calculated by multiplying along branches of a tree diagram
Two-way table — a table used to organise data or outcomes from two categories or events, showing frequencies or probabilities
Core concepts
Sample space diagrams
Sample space diagrams display all possible outcomes when two events occur simultaneously. They're particularly useful when outcomes can be listed systematically.
Creating a sample space diagram:
- List all outcomes of the first event along one axis
- List all outcomes of the second event along the other axis
- Complete the grid showing all combined outcomes
- Count favourable outcomes and divide by total outcomes
Example: Rolling two dice
Create a 6×6 grid with first die results (1-6) on one axis and second die results (1-6) on the other. Each cell shows the sum or the pair of results, giving 36 equally likely outcomes.
This method works best when:
- Both events have a manageable number of outcomes
- Outcomes are equally likely
- You need to visualise all possibilities
Two-way tables for combined events
Two-way tables organise frequencies or probabilities from two categorical variables. Unlike sample space diagrams, these often contain data from surveys or experiments.
Using two-way tables:
- Identify what each row and column represents
- Calculate row and column totals
- Add probabilities or frequencies as required
- Use the totals to find probabilities (frequency ÷ total)
Example structure:
| Event A | Not Event A | Total | |
|---|---|---|---|
| Event B | n(A and B) | n(B only) | n(B) |
| Not Event B | n(A only) | n(neither) | n(not B) |
| Total | n(A) | n(not A) | n(total) |
Two-way tables help you find:
- P(A and B) — the intersection of both events
- P(A or B) — the union (remember to avoid double-counting)
- Conditional probabilities like P(A|B)
Tree diagrams for sequential events
Tree diagrams are essential for calculating probabilities of sequential events, especially when probabilities change between stages (dependent events).
Constructing tree diagrams:
- Draw branches for all outcomes of the first event
- Label each branch with its probability
- From each first-stage outcome, draw branches for the second event
- Label these with appropriate probabilities (check if events are dependent)
- Ensure probabilities at each branching point sum to 1
Key rules:
- Multiply along branches to find the probability of a specific path
- Add across paths to find the probability of multiple scenarios leading to the same result
- All final probabilities must sum to 1
For independent events: second-stage probabilities remain constant across all branches
For dependent events: second-stage probabilities change based on the first outcome (common with "without replacement" scenarios)
Independent events
When events are independent, one outcome doesn't influence the other. This affects how you calculate combined probabilities.
Testing for independence:
Events A and B are independent if: P(A and B) = P(A) × P(B)
Examples of independent events:
- Tossing a coin twice
- Rolling a die and spinning a spinner
- Selecting items with replacement
Calculation method:
Simply multiply the individual probabilities: P(A and B) = P(A) × P(B)
This principle extends to more than two events: P(A and B and C) = P(A) × P(B) × P(C)
Dependent events and conditional probability
Dependent events require careful attention to how earlier outcomes affect later probabilities. This commonly appears in "without replacement" contexts.
Conditional probability notation:
P(A|B) means "the probability of A given that B has occurred"
For dependent events:
P(A and B) = P(A) × P(B|A)
This reads as: "probability of A, then probability of B given A has happened"
Classic example: Drawing two cards without replacement
- First draw: P(red card) = 26/52
- Second draw: P(red card|first was red) = 25/51
- P(two red cards) = (26/52) × (25/51)
The denominator decreases because the first card isn't replaced, and the numerator for the second draw depends on what happened first.
Mutually exclusive events and "or" probabilities
When calculating the probability that one event OR another occurs, you must consider whether they can happen simultaneously.
For mutually exclusive events:
P(A or B) = P(A) + P(B)
Events are mutually exclusive if they cannot both occur. Examples:
- Rolling a 2 or rolling a 5 on one die
- Drawing a king or drawing a queen (single draw)
For non-mutually exclusive events:
P(A or B) = P(A) + P(B) - P(A and B)
You subtract P(A and B) to avoid counting outcomes twice. Examples:
- Drawing a red card or drawing a king (red kings counted in both)
- Being in Year 10 or studying French (some Year 10s study French)
Exam tip: Always check whether events can occur together before adding probabilities.
Worked examples
Example 1: Tree diagram with dependent events
Question: A bag contains 5 red counters and 3 blue counters. Sarah picks two counters at random without replacement. Calculate the probability that both counters are the same colour. (4 marks)
Solution:
First, draw a tree diagram:
Red (4/7) → Red, Red: (5/8)×(4/7) = 20/56
Red (5/8)
Blue (3/7) → Red, Blue: (5/8)×(3/7) = 15/56
Start
Red (5/7) → Blue, Red: (3/8)×(5/7) = 15/56
Blue (3/8)
Blue (2/7) → Blue, Blue: (3/8)×(2/7) = 6/56
[1 mark for correct tree diagram structure]
First draw: P(Red) = 5/8, P(Blue) = 3/8
Second draw depends on first:
- If first was red: P(Red) = 4/7, P(Blue) = 3/7
- If first was blue: P(Red) = 5/7, P(Blue) = 2/7
[1 mark for correct probabilities showing dependence]
P(both same colour) = P(Red and Red) + P(Blue and Blue)
P(Red and Red) = (5/8) × (4/7) = 20/56
P(Blue and Blue) = (3/8) × (2/7) = 6/56
[1 mark for correct calculations]
P(both same) = 20/56 + 6/56 = 26/56 = 13/28
[1 mark for final answer]
Example 2: Two-way table with probabilities
Question: The table shows information about 80 students and whether they study Music or Drama.
| Study Music | Don't study Music | Total | |
|---|---|---|---|
| Study Drama | 15 | 22 | 37 |
| Don't study Drama | 18 | 25 | 43 |
| Total | 33 | 47 | 80 |
a) A student is selected at random. Find P(studies Music and Drama). (1 mark) b) Are studying Music and studying Drama independent events? Show your working. (3 marks)
Solution:
a) P(Music and Drama) = 15/80 = 3/16 [1 mark]
b) For independence, we need: P(M and D) = P(M) × P(D)
P(Music) = 33/80 P(Drama) = 37/80
P(M) × P(D) = (33/80) × (37/80) = 1221/6400 [1 mark for calculating product]
Converting to compare: 15/80 = 1200/6400
Since 1200/6400 ≠ 1221/6400, the events are not independent. [2 marks: 1 for comparison, 1 for correct conclusion]
Alternative method: P(M|D) = 15/37 and P(M) = 33/80 = 15.15.../80 Since 15/37 ≠ 33/80, not independent.
Example 3: Sample space diagram
Question: Two fair four-sided spinners are spun. Spinner A has sections numbered 1, 2, 3, 4. Spinner B has sections numbered 2, 3, 4, 5. The score is found by adding the two numbers.
a) Complete the sample space diagram. (2 marks) b) Find the probability of scoring 7 or more. (2 marks)
Solution:
a) Sample space diagram:
| + | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 1 | 3 | 4 | 5 | 6 |
| 2 | 4 | 5 | 6 | 7 |
| 3 | 5 | 6 | 7 | 8 |
| 4 | 6 | 7 | 8 | 9 |
[2 marks for all 16 outcomes correct; 1 mark if 12-15 correct]
b) Outcomes of 7 or more: 7, 7, 7, 8, 8, 9 = 6 outcomes
Total possible outcomes = 16
P(7 or more) = 6/16 = 3/8 [2 marks: 1 for correct count, 1 for probability]
Common mistakes and how to avoid them
Forgetting to adjust probabilities in dependent events. When sampling without replacement, denominators AND numerators change. After removing one item, recalculate both values for the next stage.
Adding when you should multiply, or vice versa. Remember: multiply along branches (AND), add across different paths (OR). If you want "A and then B," multiply. If you want "A or B" (different scenarios), add.
Not checking if events are mutually exclusive before using P(A or B) = P(A) + P(B). This formula only works when events cannot happen together. Otherwise, use P(A or B) = P(A) + P(B) - P(A and B).
Incorrectly assuming events are independent. Test independence using P(A and B) = P(A) × P(B), or check if P(A|B) = P(A). Don't assume without checking, especially with "without replacement" scenarios.
Misreading two-way tables. Carefully identify what rows and columns represent. The intersection of a row and column gives you "A and B," not "A or B." Always check totals add correctly.
Not simplifying fractions or giving probabilities in the wrong form. Examiners often require fractions in simplest form unless the question specifies otherwise. Check the question for required format (fraction, decimal, percentage).
Exam technique for "Probability: Combined Events"
Command word "calculate" or "find" requires clear working shown. Set out tree diagrams neatly or create sample space diagrams systematically. Marks are awarded for method even if your final answer is wrong, so show every step.
Use the correct probability format. Unless specified, give probabilities as fractions in their simplest form. Convert decimals only when instructed. Never leave answers as ratios (3:5 instead of 3/8 is incorrect).
Check probabilities sum to 1. At each branching point in a tree diagram, probabilities must total 1. This provides a quick accuracy check and prevents careless errors.
Answer the question being asked. Questions may ask for P(at least one), P(exactly two), P(both), or P(neither). These require different calculations, so highlight the key words and ensure your final answer matches what's requested.
Quick revision summary
Combined events involve calculating probabilities when two or more events occur together or sequentially. Use sample space diagrams for systematic outcomes, two-way tables for categorical data, and tree diagrams for sequential events. Remember to multiply along branches and add across paths. Independent events satisfy P(A and B) = P(A) × P(B); dependent events require adjusted probabilities. Mutually exclusive events cannot occur simultaneously, so P(A or B) = P(A) + P(B). Always show clear working and simplify final answers.