Combined Events and Conditional ProbabilityAQA GCSE Maths: Revision notes

Section 1

What are independent and dependent events?

Independent events are events where the outcome of one event does not affect the probability of the other. Dependent events are events where the outcome of the first event changes the probability of the second event occurring.

Event Type	Definition	Example
Independent	Outcome of event A has no effect on event B	Rolling a dice twice; flipping a coin and rolling a dice
Dependent	Outcome of event A affects the probability of event B	Drawing cards from a deck without replacement; selecting items from a limited set

When drawing items without replacement, the total number of items decreases, making subsequent probabilities dependent on earlier outcomes. When drawing with replacement, probabilities remain constant, indicating independence.

Key termsindependent eventsdependent eventswithout replacementwith replacement

Think of it like this

Think of independent events like two separate lottery draws with no connection between them. Dependent events are like taking sweets from a bag—once you take one, there's fewer left, so your next choice is affected.

Common mistake

Students often treat dependent events as independent. Always check: does the first outcome change the total number of possibilities for the second event? If yes, it's dependent.

Section 2

How do you calculate probabilities of independent events?

For independent events, use the multiplication rule:

P(A and B) = P(A) × P(B)

This extends to three or more independent events:

P(A and B and C) = P(A) × P(B) × P(C)

The key principle: multiply the individual probabilities together to find the probability of all events occurring.

Step-by-step process:

Identify each individual event
Calculate the probability of each event separately
Multiply all probabilities together
Simplify your final answer as a fraction or decimal

Important: This rule only applies when events are independent. If events are dependent, the probabilities change after each outcome.

Key termsmultiplication ruleP(A and B)independent events

Example

A fair coin is flipped and a fair dice is rolled. Find P(heads and rolling a 4). P(heads) = 1/2, P(rolling a 4) = 1/6. P(heads and 4) = 1/2 × 1/6 = 1/12.

Exam tip

Examiners want to see the multiplication rule clearly stated and applied. Write 'P(A and B) = P(A) × P(B)' explicitly in your working to gain full marks.

Section 3

How do tree diagrams help with combined events?

Tree diagrams are visual representations of all possible outcomes in combined events. They clearly show:

Each possible outcome at each stage
The probability of each branch
How events are connected (independent or dependent)

Constructing a tree diagram:

Draw branches from a starting point for the first event
Label each branch with its probability
From the end of each first-event branch, draw branches for the second event
Label second-event branches with their probabilities (remembering to update these for dependent events)
Multiply along branches to find combined probabilities

Reading tree diagrams:

Multiply along a path to find the probability of a specific sequence of outcomes (e.g., "heads then tails")
Add different paths if they lead to the same overall outcome (e.g., "at least one heads")

Tree diagrams work for both independent and dependent events. For dependent events, update the numbers on later branches based on earlier outcomes.

Key termstree diagrambranchespathcombined probability

Exam tip

For two-event tree diagrams, always check your probabilities on each branch sum to 1. For dependent events, ensure second-event probabilities reflect updated totals.

Example

Drawing two cards without replacement from {Red, Red, Blue, Green}. First card probability branches: P(Red) = 2/4, P(Blue) = 1/4, P(Green) = 1/4. Second branches adjust: if Red drawn first, P(Red from remaining) = 1/3 (not 2/4).

Section 4

What is conditional probability and how do you apply it?

Conditional probability is the probability of an event occurring given that another event has already occurred. It's written as P(A | B), meaning "the probability of A given that B has occurred."

Conditional probability typically arises in dependent event situations, where knowing the outcome of one event changes the probability of another.

Using tree diagrams for conditional probability:

The probabilities on the second branch depend on what happened first
These are conditional probabilities—they only apply given the first outcome
Always update total numbers and available options when calculating conditional probabilities

Using Venn diagrams for conditional probability:

Venn diagrams show overlapping regions representing events
To find P(A | B), focus only on the region where B occurs
Calculate: P(A | B) = (Number in both A and B) / (Total number in B)
This isolates the "given B" condition by making B the new "total"

Two-way tables and conditional probability:

Two-way tables organise data about two variables
To find a conditional probability, identify the relevant row or column (the condition)
Calculate the probability within that row/column only

Key termsconditional probabilityP(A | B)given thatVenn diagramtwo-way table

Example

A two-way table shows 120 students: 60 study maths, 80 study English, 40 study both. Find P(Maths | English). Given English is studied: P(Maths | English) = 40/80 = 1/2. Use only the 'English' column total as the denominator.

Common mistake

Students often forget to update denominators for conditional probability. P(A | B) must use the total number in B as the denominator, not the original total.

Section 5

How do you use the addition law with combined events?

The addition law calculates the probability of at least one event occurring:

P(A or B) = P(A) + P(B) – P(A and B)

The subtraction of P(A and B) is crucial: it prevents double-counting the overlap where both events occur.

When to use the addition law:

Finding probability of "A or B" (or both)
Events are mutually exclusive (can't both happen): P(A or B) = P(A) + P(B)
Events overlap: use the full formula with the subtraction term

Comparing mutually exclusive vs overlapping events:

Situation	Formula	Example
Mutually exclusive (can't both occur)	P(A or B) = P(A) + P(B)	Rolling 1 or rolling 2 on a dice
Overlapping (can both occur)	P(A or B) = P(A) + P(B) – P(A and B)	Student studies maths or English (or both)

Connection to tree diagrams and Venn diagrams:

Use tree diagrams to identify all paths leading to "A or B" outcomes, then add these probabilities
Use Venn diagrams to visualise overlaps and identify P(A and B) clearly
With two-way tables, add relevant cells but subtract any overlap to avoid double-counting

Key termsaddition lawP(A or B)mutually exclusiveoverlapdouble-counting

Example

P(Red) = 0.3, P(Blue) = 0.2, P(Red and Blue) = 0.05. Find P(Red or Blue). Using the addition law: P(Red or Blue) = 0.3 + 0.2 – 0.05 = 0.45.

Exam tip

Always check whether events can overlap. If they're mutually exclusive, P(A and B) = 0, so the formula simplifies to P(A) + P(B). Examiners expect you to identify this and simplify accordingly.

Section 6

How do two-way tables help calculate probabilities?

Two-way tables organise data about two variables in a grid format, making it easy to calculate both simple and conditional probabilities.

Structure of a two-way table:

Rows represent one variable (e.g., A/not A)
Columns represent another variable (e.g., B/not B)
Cells contain frequencies or counts
Totals row and column show the sum for each category

Calculating probabilities from two-way tables:

Simple probability: P(event) = (Frequency for that event) / (Total frequency)
Combined probability: P(A and B) = (Frequency in both A and B cell) / (Total frequency)
Conditional probability: P(A | B) = (Frequency in both A and B) / (Total frequency in B)

Key principle: Always identify which frequencies to use based on what the question asks. For conditional probabilities, the condition determines which row or column totals become your denominator.

Two-way tables vs tree diagrams:

Two-way tables are better for summarising data and conditional probability from known frequencies
Tree diagrams are better for sequential events and visualising independent/dependent changes
Both can solve the same problem; choose based on what's given in the question

Key termstwo-way tablerowscolumnsfrequenciestotals

Example

A two-way table has 100 people: 40 like coffee (20 also like tea), 60 don't like coffee (30 like tea). Find P(Coffee and Tea). P(Coffee and Tea) = 20/100 = 0.2.

Exam tip

Always include the totals row and column when setting up a two-way table. This makes probability calculations clearer and helps prevent errors in identifying denominators.

Must Know

Independent events: P(A and B) = P(A) × P(B). Use this for events that don't affect each other, such as rolling dice twice or coin flips.
Dependent events: Probabilities change after the first outcome (e.g., drawing without replacement). Always update totals and available options on tree diagram branches.
Tree diagrams: Multiply along a single path for one specific sequence; add different paths to find "at least one" or alternative outcomes. Label all branches clearly with probabilities.
Conditional probability: P(A | B) means "probability of A given B has occurred." Use the frequency of both divided by the frequency of B (from a two-way table), or focus on the relevant branch of a tree diagram.
Addition law: P(A or B) = P(A) + P(B) – P(A and B). Subtract P(A and B) to avoid double-counting overlaps. For mutually exclusive events, this term equals zero.
Two-way tables: Extract simple probabilities by dividing cell frequencies by total; use conditional probabilities by dividing the intersection by the row/column total (the condition). Both tree diagrams and two-way tables solve combined event problems; choose based on whether the question emphasises sequence or summarised data.

That's the notes covered.

Carry on to the next subtopic.