A bag has 3 red and 2 blue balls. One red ball is removed. What is the probability the next ball is red?
Total and reds both change.
Now 2 red out of 4 total → 1/2.
Conditional Probability
Conditional probability considers how the likelihood of an event changes when another event has already occurred. It builds on ideas from basic probability and is often explored using Venn diagrams and tree diagrams.
Overview
Conditional probability is the probability of one event happening given that another event has already happened.
\( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)
The key idea is that once you are told event B has happened, you no longer use the original whole sample space. You only work inside event B.
What you should understand after this topic
- Understand what 'given that' means in probability
- Recognise how the sample space is reduced
- Use conditional probability notation correctly
- Read conditional probability from diagrams and tables
- Avoid confusing intersection with conditional probability
Key Definitions
Conditional Probability
The probability of an event happening given that another event has already happened.
Given That
A phrase meaning you already know some extra information.
Sample Space
The full set of possible outcomes.
Restricted Sample Space
The smaller set of outcomes left after extra information is given.
Intersection
\(A \cap B\) means outcomes that are in both \(A\) and \(B\).
Notation \(P(A \mid B)\)
The probability of \(A\) given \(B\).
Key Rules
Main formula
\( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)
New denominator
When given \(B\), the denominator becomes \(P(B)\), not the full sample space.
Intersection first
The numerator is the part where both events happen together.
Look inside the condition
If the question says “given \(B\)”, work only inside event \(B\).
Quick Interpretation Check
\(P(A \cap B)\)
Probability of both events happening.
\(P(A \mid B)\)
Probability of \(A\) when you already know \(B\) happened.
\(P(B)\)
The size of the restricted sample space in the denominator.
Important idea
Conditional probability changes what counts as “the whole”.
How to Solve
What does “given that” mean?
In standard probability, you consider all possible outcomes. In conditional probability, you are given extra information that reduces the sample space.
For example, if a card is chosen and you are told it is a heart, you no longer consider all 52 cards. You only consider the 13 hearts.
Exam tip: Always identify the new reduced sample space first.
The main idea
Conditional probability focuses only on outcomes that satisfy the given condition.
\( \text{Probability wanted} = \frac{\text{favourable outcomes inside the condition}}{\text{all outcomes inside the condition}} \)
Formula
\( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)
This builds on ideas from probability basics and scale.
Meaning of the formula
The numerator represents outcomes where both \(A\) and \(B\) occur. The denominator represents all outcomes in event \(B\).
Using Venn diagrams
In a Venn diagram, conditional probability means you focus only on the region inside the condition.
This method connects directly to Venn diagrams.
For \(P(A \mid B)\)
Focus only on circle \(B\).
Favourable outcomes
Count the overlap \(A \cap B\) within that region.
Using two-way tables
In a two-way table, the condition tells you which row or column becomes the new total.
This is closely linked to two-way tables.
Example idea
Find the probability that a student is left-handed given that the student is male. Only the male row or column is used.
Using tree diagrams
In tree diagrams, probabilities on later branches may depend on earlier outcomes. This is also conditional probability.
Tree diagrams are covered in tree diagrams.
Example
If a ball is taken from a bag and not replaced, the second probability depends on the first outcome.
Conditional probability and multiplication
\( P(A \cap B) = P(A) \times P(B \mid A) \)
This formula is used when finding the probability of two dependent events happening together.
For events that cannot happen at the same time, see mutually exclusive events.
Conditional probability and multiplication
\( P(A \cap B) = P(A) \times P(B \mid A) \)
This formula is used when finding the probability of two dependent events happening together.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on conditional probability and probability without replacement.
Edexcel
A bag contains 7 orange sweets and 3 white sweets. A sweet is taken at random and eaten. A second sweet is then taken at random and eaten.
Find the probability that both sweets are orange.
Edexcel
There are only n red counters and 2 blue counters in a bag. A counter is taken at random and not replaced. A second counter is then taken at random.
Show that the probability of taking two red counters is \( \frac{n(n-1)}{(n+2)(n+1)} \).
Edexcel
A box contains 5 green counters, 4 yellow counters and 3 red counters. A counter is taken at random and not replaced. A second counter is then taken at random.
Given that the first counter is green, find the probability that the second counter is yellow.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on tree diagrams, dependent events and conditional probability.
AQA
There are 6 red counters and 4 blue counters in a bag. Two counters are taken at random without replacement.
Complete a probability tree diagram for this information.
AQA
There are 8 red counters and 5 yellow counters in a bag. A counter is taken at random and not replaced. A second counter is then taken at random.
Find the probability that the second counter is yellow, given that the first counter was red.
AQA
A teacher chooses 2 students at random from a class of 25 students.
Work out the number of different pairs of students that can be chosen.
AQA
A student says that choosing two students is the same as choosing one student twice with replacement.
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising probability reasoning, conditional outcomes and multi-step dependent events.
OCR
There are 4 red counters, 3 yellow counters and 2 green counters in a bag. Three counters are taken at random without replacement.
Work out the probability that all three counters are yellow, red and green in any order.
OCR
There are 5 red counters, 4 blue counters and 1 green counter in a bag. Two counters are taken at random without replacement.
Given that the first counter is not green, find the probability that the second counter is blue.
OCR
A bag contains 4 red counters, 3 yellow counters and 1 blue counter. Three counters are taken at random without replacement.
Work out the probability that there are more yellow counters than red counters left in the bag.
Exam Checklist
Step 1
Underline the words “given that”.
Step 2
Work out the new restricted sample space.
Step 3
Find the favourable outcomes inside that restricted set.
Step 4
Write the fraction carefully and simplify if needed.
Most common exam mistakes
Wrong denominator
Using the total number of outcomes instead of the given event.
Wrong overlap
Choosing the whole event instead of the shared part.
Table mistake
Using the wrong row or column total.
Tree mistake
Ignoring that probabilities change after earlier events.
Common Mistakes
These are common mistakes students make when working with conditional probability in GCSE Maths.
Using the original total instead of the restricted total
Incorrect
A student uses the full sample size even after a condition is given.
Correct
When a condition is applied, the sample space is reduced. You must use the restricted total that satisfies the condition.
Confusing \(P(A \cap B)\) with \(P(A \mid B)\)
Incorrect
A student treats intersection and conditional probability as the same.
Correct
\(P(A \cap B)\) means both events happen, while \(P(A \mid B)\) means the probability of A given that B has already occurred. The denominator changes for conditional probability.
Choosing the wrong row or column in a table
Incorrect
A student selects values from the wrong part of a two-way table.
Correct
Carefully identify which condition is being applied and select the correct row or column that matches it.
Ignoring the effect of “given”
Incorrect
A student calculates probability without adjusting for the condition.
Correct
The word “given” means the sample space has changed. Only outcomes that satisfy the condition should be considered.
Assuming replacement when there is none
Incorrect
A student treats events as independent when items are not replaced.
Correct
Without replacement, probabilities change after each selection. Always adjust the totals accordingly.
Try It Yourself
Practise calculating probabilities involving dependent events.
Foundation
Foundation Practice
Understand conditional probability in simple contexts.
A bag has 4 red and 1 blue ball. One red is removed. Find the probability the next ball is red.
Update totals.
3 red out of 4 total → 3/4.
What does 'given that' mean in probability?
Information is updated.
'Given that' means we are working with new information.
A bag has 5 balls. 2 are red. Given that one red is removed, how many balls remain?
Remove one ball.
5 − 1 = 4 balls remain.
A bag has 3 red and 3 blue balls. Given that a red ball was picked first, what is the probability the second ball is blue?
Update totals.
Now 3 blue out of 5 total → 3/5.
A bag has 6 balls: 2 red, 4 blue. Given that a blue was removed, find probability of picking a red next.
Update total.
2 red out of 5 total → 2/5.
Why does probability change after removing an item?
Sample space changes.
Removing items changes total outcomes.
A bag has 3 red and 2 blue balls. Given that a blue was picked first, how many balls are left?
Remove one ball.
5 − 1 = 4 balls remain.
If an event depends on a previous outcome, it is called:
Depends on previous result.
Dependent events rely on previous outcomes.
A bag has 4 red and 4 blue balls. Given one red is removed, find probability of picking blue next.
Update total.
4 blue out of 7 total → 4/7.
Higher
Higher Practice
Calculate conditional probabilities using fractions and reasoning.
A bag has 5 red and 3 blue balls. Two are picked without replacement. What is the probability both are red?
Multiply adjusted probabilities.
5/8 × 4/7 = 20/56 = 5/14.
A bag has 6 balls: 3 red, 3 blue. Two are picked without replacement. Find probability of red then blue.
Multiply adjusted probabilities.
3/6 × 3/5 = 9/30 = 3/10.
What is the key difference between independent and dependent events?
Think about changing totals.
Dependent events change probabilities.
A bag has 4 red and 2 blue balls. Two are picked without replacement. Find probability of at least one red.
Use complement.
P(no red) = blue-blue = 2/6 × 1/5 = 2/30 = 1/15 → 1 − 1/15 = 14/15.
Why is the complement method useful?
Easier than listing cases.
Complement simplifies problems.
A bag has 5 balls: 2 red, 3 blue. Two are picked without replacement. Find probability both are blue.
Multiply adjusted probabilities.
3/5 × 2/4 = 6/20 = 3/10.
A student forgets to change probabilities after removing a ball. What is wrong?
Think dependency.
Dependent events require updated probabilities.
A bag has 3 red and 2 blue balls. Two are picked without replacement. Find probability of one red and one blue (any order).
Add both cases.
RB = 3/5 × 2/4 = 6/20 = 3/10, BR = 2/5 × 3/4 = 6/20 = 3/10 → total = 3/5.
Which situation involves dependent events?
Does outcome affect next?
Without replacement → dependent.
A bag has 7 balls: 4 red, 3 blue. Two are picked without replacement. Find probability both are red.
Multiply adjusted probabilities.
4/7 × 3/6 = 12/42 = 2/7.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is conditional probability?
Probability given that another event has occurred.
When do I use it?
When events are dependent.
What tools help?
Tree diagrams or Venn diagrams.