11-15-2025, 04:31 PM
Chapter 7 — Conditional Probability
Conditional probability is one of the hardest topics in GCSE maths — and even many adults
don’t understand it.
But the idea becomes clear once you learn the right way to think about it.
In this chapter we break conditional probability into simple, approachable steps.
---
7.1 What Conditional Probability Means
Conditional probability is when:
You already know something has happened,
and you want the probability of a second event GIVEN this information.
Example:
A bag has 5 red, 3 blue, 2 green sweets.
If someone tells you:
"A red sweet has already been chosen,"
that changes the probabilities for the next pick.
We are no longer considering the whole bag — we are conditioning on new information.
---
7.2 The Symbol “Given that”
Conditional probability is written like:
P(A | B)
Which is read as:
"The probability of A GIVEN B has already happened."
Example:
P(blue | red was chosen first)
This means:
Probability of blue, knowing red has been removed.
---
7.3 The Big Idea
Conditional probability =
Probability based on a smaller world.
The moment you are given new information,
the sample space (the total possibilities) changes.
Example:
A class has 10 boys and 10 girls.
If someone tells you:
"The student chosen is a girl,"
the probability they have long hair is now:
(number of long-haired girls) / (number of girls)
NOT:
(number of long-haired students) / (total students)
---
7.4 Simple Example
A bag has:
• 4 red
• 3 blue
• 3 yellow
Someone picks a sweet and tells you it is NOT red.
So now the only possibilities are:
blue or yellow
New total = 3 + 3 = 6
Q: What is P(blue | not red)?
Blue = 3
New total = 6
So:
P(blue | not red) = 3/6 = 1/2
This demonstrates how information changes probability.
---
7.5 Conditional Probability with No Replacement
This is where most exam questions appear.
Example:
A bag has 6 green and 4 red sweets.
Pick one and do NOT put it back.
Find:
P(red second | green first)
Step 1 — Remove a green
Remaining:
5 green, 4 red → total = 9
Step 2 — Red second:
4/9
So:
P(red second | green first) = 4/9
---
7.6 A Common Exam Question
A box has 3 blue and 7 yellow pens.
One pen is chosen at random and NOT replaced.
Find:
P(blue second | yellow first).
Step 1 — Remove a yellow:
Blue = 3
Yellow = 6
Total = 9
Step 2 — Probability blue second:
3/9 = 1/3
---
7.7 Why Conditional Probability Is Hard
Because students forget:
You are no longer working with the original total.
The new information SHRINKS the sample space.
---
7.8 Worked Examples
Example 1
A class has 15 students:
• 9 girls
• 6 boys
5 girls have long hair, 4 do not.
Find:
P(long hair | girl)
Girls with long hair = 5
Total girls = 9
P = 5/9
---
Example 2
A bag has 2 red, 2 blue, 1 green.
One sweet is chosen and it is revealed to be blue.
Find:
P(green | blue was chosen)
After knowing the chosen sweet is blue:
The next pick is from the FULL bag (because the first is not replaced yet).
Blue removed:
Remaining:
2 red, 1 blue, 1 green → total = 4
Probability green:
1/4
---
Example 3
A deck has 52 cards.
Find:
P(ace | the card chosen is a spade)
Number of spades = 13
Number of aces in spades = 1
Conditional sample space = 13 cards
Desired = 1 ace
P = 1/13
---
Example 4
A group has 12 people:
• 5 are left-handed
• 7 are right-handed
3 of the left-handed people wear glasses.
2 of the right-handed people wear glasses.
Find:
P(glasses | right-handed)
Right-handed = 7
Right-handed with glasses = 2
P = 2/7
---
7.9 Common Pitfalls
Mistake 1: Using original totals
Conditional probability ALWAYS uses a smaller sample space.
Mistake 2: Forgetting order matters
(because earlier events change the totals)
Mistake 3: Mixing up independent and dependent events
Mistake 4: Thinking conditional means multiply
Sometimes yes — sometimes no.
It depends on the question and tree diagrams (next chapter).
---
7.10 Your Turn
1. A bag has 5 black and 5 white stones.
One is removed and revealed to be black.
Find P(white | black was removed).
2. A class has 8 boys and 12 girls.
6 girls have pets.
Find P(pet | girl).
3. A jar has 4 red, 3 blue, 1 yellow.
A sweet is chosen and revealed NOT to be red.
Find P(yellow | not red).
4. A deck has 52 cards.
Find P(queen | the card is a face card).
(Hint: face cards are J, Q, K)
5. A survey shows:
• 40 people drink tea
• 20 of those also drink coffee
Find P(coffee | tea).
---
Chapter Summary
• Conditional probability = probability with new information
• The sample space SHRINKS
• P(A | B) means "probability of A given B"
• Used constantly in no-replacement problems
• Used heavily in tree diagrams and combined events
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Conditional probability is one of the hardest topics in GCSE maths — and even many adults
don’t understand it.
But the idea becomes clear once you learn the right way to think about it.
In this chapter we break conditional probability into simple, approachable steps.
---
7.1 What Conditional Probability Means
Conditional probability is when:
You already know something has happened,
and you want the probability of a second event GIVEN this information.
Example:
A bag has 5 red, 3 blue, 2 green sweets.
If someone tells you:
"A red sweet has already been chosen,"
that changes the probabilities for the next pick.
We are no longer considering the whole bag — we are conditioning on new information.
---
7.2 The Symbol “Given that”
Conditional probability is written like:
P(A | B)
Which is read as:
"The probability of A GIVEN B has already happened."
Example:
P(blue | red was chosen first)
This means:
Probability of blue, knowing red has been removed.
---
7.3 The Big Idea
Conditional probability =
Probability based on a smaller world.
The moment you are given new information,
the sample space (the total possibilities) changes.
Example:
A class has 10 boys and 10 girls.
If someone tells you:
"The student chosen is a girl,"
the probability they have long hair is now:
(number of long-haired girls) / (number of girls)
NOT:
(number of long-haired students) / (total students)
---
7.4 Simple Example
A bag has:
• 4 red
• 3 blue
• 3 yellow
Someone picks a sweet and tells you it is NOT red.
So now the only possibilities are:
blue or yellow
New total = 3 + 3 = 6
Q: What is P(blue | not red)?
Blue = 3
New total = 6
So:
P(blue | not red) = 3/6 = 1/2
This demonstrates how information changes probability.
---
7.5 Conditional Probability with No Replacement
This is where most exam questions appear.
Example:
A bag has 6 green and 4 red sweets.
Pick one and do NOT put it back.
Find:
P(red second | green first)
Step 1 — Remove a green
Remaining:
5 green, 4 red → total = 9
Step 2 — Red second:
4/9
So:
P(red second | green first) = 4/9
---
7.6 A Common Exam Question
A box has 3 blue and 7 yellow pens.
One pen is chosen at random and NOT replaced.
Find:
P(blue second | yellow first).
Step 1 — Remove a yellow:
Blue = 3
Yellow = 6
Total = 9
Step 2 — Probability blue second:
3/9 = 1/3
---
7.7 Why Conditional Probability Is Hard
Because students forget:
You are no longer working with the original total.
The new information SHRINKS the sample space.
---
7.8 Worked Examples
Example 1
A class has 15 students:
• 9 girls
• 6 boys
5 girls have long hair, 4 do not.
Find:
P(long hair | girl)
Girls with long hair = 5
Total girls = 9
P = 5/9
---
Example 2
A bag has 2 red, 2 blue, 1 green.
One sweet is chosen and it is revealed to be blue.
Find:
P(green | blue was chosen)
After knowing the chosen sweet is blue:
The next pick is from the FULL bag (because the first is not replaced yet).
Blue removed:
Remaining:
2 red, 1 blue, 1 green → total = 4
Probability green:
1/4
---
Example 3
A deck has 52 cards.
Find:
P(ace | the card chosen is a spade)
Number of spades = 13
Number of aces in spades = 1
Conditional sample space = 13 cards
Desired = 1 ace
P = 1/13
---
Example 4
A group has 12 people:
• 5 are left-handed
• 7 are right-handed
3 of the left-handed people wear glasses.
2 of the right-handed people wear glasses.
Find:
P(glasses | right-handed)
Right-handed = 7
Right-handed with glasses = 2
P = 2/7
---
7.9 Common Pitfalls
Mistake 1: Using original totals
Conditional probability ALWAYS uses a smaller sample space.
Mistake 2: Forgetting order matters
(because earlier events change the totals)
Mistake 3: Mixing up independent and dependent events
Mistake 4: Thinking conditional means multiply
Sometimes yes — sometimes no.
It depends on the question and tree diagrams (next chapter).
---
7.10 Your Turn
1. A bag has 5 black and 5 white stones.
One is removed and revealed to be black.
Find P(white | black was removed).
2. A class has 8 boys and 12 girls.
6 girls have pets.
Find P(pet | girl).
3. A jar has 4 red, 3 blue, 1 yellow.
A sweet is chosen and revealed NOT to be red.
Find P(yellow | not red).
4. A deck has 52 cards.
Find P(queen | the card is a face card).
(Hint: face cards are J, Q, K)
5. A survey shows:
• 40 people drink tea
• 20 of those also drink coffee
Find P(coffee | tea).
---
Chapter Summary
• Conditional probability = probability with new information
• The sample space SHRINKS
• P(A | B) means "probability of A given B"
• Used constantly in no-replacement problems
• Used heavily in tree diagrams and combined events
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive