11-15-2025, 05:25 PM
Chapter 20 — Mastery Test
This is the final assessment of the course.
It contains a mixture of:
• exam-level questions
• real-world problems
• multi-step probability reasoning
• data handling
• distributions
• expected value
• traps designed to test deep understanding
No calculators are strictly required — but clear reasoning is essential.
Take your time, show your thinking, and treat this as a full-length paper.
---
20.1 Section A — Core Probability Skills
Q1. A bag contains:
• 12 red counters
• 8 blue counters
• 10 yellow counters
One counter is chosen at random.
a) Find the probability it is blue.
b) Find the probability it is NOT yellow.
c) A second counter is drawn WITHOUT replacement.
Find the probability both counters are red.
---
Q2. A biased coin lands on Heads with probability 0.37.
It is flipped 5 times.
a) What is the probability of getting exactly 2 Heads?
b) What is the probability of getting at least 1 Tail?
---
20.2 Section B — Combined & Conditional Probability
Q3 — The Two-School Problem
School A has 600 students.
School B has 900 students.
In School A:
• 40% play a musical instrument
In School B:
• 25% play a musical instrument
A student is chosen at random from the combined population.
Find the probability that they:
a) play a musical instrument
b) are from School A given that they **play an instrument**
(Use conditional probability.)
---
Q4 — The Machine Failure Problem
A factory has two machines:
Machine 1: produces 70% of all items
Machine 2: produces 30% of all items
Fault rates:
• Machine 1: 3%
• Machine 2: 8%
A random item is tested and found to be faulty.
Find the probability it came from Machine 2.
---
20.3 Section C — Data Handling & Statistics
Q5. The table shows the number of books read by students last month:
Books read (x): 0, 1, 2, 3, 4, 5
Frequency: 5, 7, 12, 6, 4, 2
Calculate:
a) the mean
b) the median
c) the mode
d) the interquartile range (IQR)
---
Q6. A dataset has:
• mean = 54
• standard deviation = 7
A value of 68 is recorded.
a) Find its Z-score.
b) Explain in words what this Z-score means.
---
Q7 — The Misleading Chart
A bar chart shows company profit increasing sharply over 4 years.
However, the Y-axis starts at £950,000 and ends at £1,020,000.
Explain clearly why this graph is misleading and describe
**two changes** that would make the graph honest.
---
20.4 Section D — Expected Value & Decision Making
Q8 — The Lucky Envelope Game
A game costs £2 to play. You pick one envelope:
• £10 (probability 0.10)
• £4 (probability 0.15)
• £1 (probability 0.50)
• £0 (probability 0.25)
a) Calculate the expected winnings.
b) Is the game fair?
c) If not, what would the prize of the £10 envelope need to be
to make the game fair?
---
20.5 Section E — Distributions
Q9 — The Exam Scores Problem
Exam marks are normally distributed with mean 62 and standard deviation 9.
Find the probability a randomly chosen student scored:
a) above 70
b) between 55 and 65
(Use Z-scores; numerical table lookup optional — focus on reasoning.)
---
Q10 — The Binomial Shooting Problem
A basketball player succeeds with probability p = 0.72 per shot.
They take 15 shots.
a) Find the probability they score exactly 12 baskets.
b) Find the probability they score at least 10 baskets.
c) Explain why the binomial distribution is the appropriate model.
---
20.6 Section F — Real-World Mastery Questions
Q11 — The Mixed Boxes Problem
Box 1: 2 red, 8 green
Box 2: 7 red, 3 green
Box 3: 4 red, 6 green
A box is chosen at random (equal chance).
A ball is drawn.
a) Find the probability the ball is red.
b) Given that a red ball was drawn, find the probability it came from Box 2.
---
Q12 — The Airport Delay Problem
Two airports are compared:
Airport X:
• 65% on time
• 35% delayed
• Average delay = 18 minutes
Airport Y:
• 80% on time
• 20% delayed
• Average delay = 42 minutes
A traveller picks an airport at random and takes a flight.
a) Find the probability their flight is delayed.
b) Find the expected delay time overall.
---
Q13 — The Sampling Trap
A survey claims:
“90% of people prefer our new drink!”
Later it is revealed the survey was done:
• outside the factory of the drink company
• with only 40 people participating
Explain **three reasons** this statistic is unreliable.
---
20.7 Section G — Final Challenge Question
Q14 — The Grand Probability Puzzle
A bag contains:
• 4 black balls
• 6 white balls
• 5 gold balls
A ball is drawn, its colour is recorded, and then it is returned.
This is done **four times**.
Find the probability that:
a) exactly one gold ball is drawn
b) at least two white balls are drawn
c) no black balls are drawn
Show all reasoning.
---
20.8 Chapter Summary
• This mastery test assesses all course topics
• Problems include combined probability, conditional probability, expected value, distributions, and real-world traps
• Questions require clear reasoning and interpretation
• Completing this test means you have full working knowledge of GCSE-level probability & statistics
• You are now prepared for more advanced work or higher-tier Lumin Archive challenges
Congratulations — you’ve completed Course 02.
You should be proud.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
This is the final assessment of the course.
It contains a mixture of:
• exam-level questions
• real-world problems
• multi-step probability reasoning
• data handling
• distributions
• expected value
• traps designed to test deep understanding
No calculators are strictly required — but clear reasoning is essential.
Take your time, show your thinking, and treat this as a full-length paper.
---
20.1 Section A — Core Probability Skills
Q1. A bag contains:
• 12 red counters
• 8 blue counters
• 10 yellow counters
One counter is chosen at random.
a) Find the probability it is blue.
b) Find the probability it is NOT yellow.
c) A second counter is drawn WITHOUT replacement.
Find the probability both counters are red.
---
Q2. A biased coin lands on Heads with probability 0.37.
It is flipped 5 times.
a) What is the probability of getting exactly 2 Heads?
b) What is the probability of getting at least 1 Tail?
---
20.2 Section B — Combined & Conditional Probability
Q3 — The Two-School Problem
School A has 600 students.
School B has 900 students.
In School A:
• 40% play a musical instrument
In School B:
• 25% play a musical instrument
A student is chosen at random from the combined population.
Find the probability that they:
a) play a musical instrument
b) are from School A given that they **play an instrument**
(Use conditional probability.)
---
Q4 — The Machine Failure Problem
A factory has two machines:
Machine 1: produces 70% of all items
Machine 2: produces 30% of all items
Fault rates:
• Machine 1: 3%
• Machine 2: 8%
A random item is tested and found to be faulty.
Find the probability it came from Machine 2.
---
20.3 Section C — Data Handling & Statistics
Q5. The table shows the number of books read by students last month:
Books read (x): 0, 1, 2, 3, 4, 5
Frequency: 5, 7, 12, 6, 4, 2
Calculate:
a) the mean
b) the median
c) the mode
d) the interquartile range (IQR)
---
Q6. A dataset has:
• mean = 54
• standard deviation = 7
A value of 68 is recorded.
a) Find its Z-score.
b) Explain in words what this Z-score means.
---
Q7 — The Misleading Chart
A bar chart shows company profit increasing sharply over 4 years.
However, the Y-axis starts at £950,000 and ends at £1,020,000.
Explain clearly why this graph is misleading and describe
**two changes** that would make the graph honest.
---
20.4 Section D — Expected Value & Decision Making
Q8 — The Lucky Envelope Game
A game costs £2 to play. You pick one envelope:
• £10 (probability 0.10)
• £4 (probability 0.15)
• £1 (probability 0.50)
• £0 (probability 0.25)
a) Calculate the expected winnings.
b) Is the game fair?
c) If not, what would the prize of the £10 envelope need to be
to make the game fair?
---
20.5 Section E — Distributions
Q9 — The Exam Scores Problem
Exam marks are normally distributed with mean 62 and standard deviation 9.
Find the probability a randomly chosen student scored:
a) above 70
b) between 55 and 65
(Use Z-scores; numerical table lookup optional — focus on reasoning.)
---
Q10 — The Binomial Shooting Problem
A basketball player succeeds with probability p = 0.72 per shot.
They take 15 shots.
a) Find the probability they score exactly 12 baskets.
b) Find the probability they score at least 10 baskets.
c) Explain why the binomial distribution is the appropriate model.
---
20.6 Section F — Real-World Mastery Questions
Q11 — The Mixed Boxes Problem
Box 1: 2 red, 8 green
Box 2: 7 red, 3 green
Box 3: 4 red, 6 green
A box is chosen at random (equal chance).
A ball is drawn.
a) Find the probability the ball is red.
b) Given that a red ball was drawn, find the probability it came from Box 2.
---
Q12 — The Airport Delay Problem
Two airports are compared:
Airport X:
• 65% on time
• 35% delayed
• Average delay = 18 minutes
Airport Y:
• 80% on time
• 20% delayed
• Average delay = 42 minutes
A traveller picks an airport at random and takes a flight.
a) Find the probability their flight is delayed.
b) Find the expected delay time overall.
---
Q13 — The Sampling Trap
A survey claims:
“90% of people prefer our new drink!”
Later it is revealed the survey was done:
• outside the factory of the drink company
• with only 40 people participating
Explain **three reasons** this statistic is unreliable.
---
20.7 Section G — Final Challenge Question
Q14 — The Grand Probability Puzzle
A bag contains:
• 4 black balls
• 6 white balls
• 5 gold balls
A ball is drawn, its colour is recorded, and then it is returned.
This is done **four times**.
Find the probability that:
a) exactly one gold ball is drawn
b) at least two white balls are drawn
c) no black balls are drawn
Show all reasoning.
---
20.8 Chapter Summary
• This mastery test assesses all course topics
• Problems include combined probability, conditional probability, expected value, distributions, and real-world traps
• Questions require clear reasoning and interpretation
• Completing this test means you have full working knowledge of GCSE-level probability & statistics
• You are now prepared for more advanced work or higher-tier Lumin Archive challenges
Congratulations — you’ve completed Course 02.
You should be proud.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive