11-15-2025, 05:23 PM
Chapter 19 — Real Exam-Style Problems
You’ve reached the chapter where we test EVERYTHING you’ve learned so far.
These questions are designed to feel like:
• GCSE Higher
• real-world tests
• logic puzzles
• mixed-topic challenges
Each problem requires you to think, combine ideas, and apply proper reasoning.
Work through them carefully — and avoid the temptation to guess.
---
19.1 Problem Set A — Ratios, Fractions & Basic Probability
Q1. A bag contains red, blue, and green counters in the ratio
3 : 5 : 2.
There are 50 counters in total.
Find how many of each colour there are.
---
Q2. A coin is biased.
It lands on heads with probability 0.62.
What is the probability of landing tails?
---
Q3. A spinner has 8 equal sections numbered 1–8.
A number is chosen at random.
Find the probability the number is:
• a multiple of 2
• greater than 5
• both (even AND > 5)
---
19.2 Problem Set B — Independent & Dependent Probability
Q4. A box contains:
• 6 black marbles
• 4 white marbles
Two marbles are drawn without replacement.
Find the probability that:
1. both are black
2. one black and one white (in any order)
---
Q5. A train is late 30% of the time.
If trains are independent day to day:
Find the probability the train is late exactly 2 times in a week (7 days).
(You may use the binomial formula or a clear reasoning method.)
---
19.3 Problem Set C — Conditional Probability
Q6. In a college:
• 40% study science
• 60% study arts
• 25% of science students play a sport
• 10% of arts students play a sport
A student is chosen at random.
They are found to play a sport.
Find the probability that they study science.
(You should use a tree diagram or conditional probability formula.)
---
19.4 Problem Set D — Tables, Frequency & Averages
Q7. A group of students were asked how many hours they revised last week.
Their results are shown:
Hours revised (h): 0, 1, 2, 3, 4
Frequency: 6, 8, 10, 5, 1
Calculate:
1. the mean
2. the median
3. the mode
4. the range
---
19.5 Problem Set E — Expected Value
Q8. A game costs £3 to play.
You draw a card at random:
• Win £10 (probability 0.15)
• Win £3 (probability 0.25)
• Win £1 (probability 0.40)
• Win £0 (probability 0.20)
Find the expected value, then determine whether the game is fair.
---
19.6 Problem Set F — Distributions (Normal & Binomial)
Q9. A machine fills bags of flour with mean 1000g and standard deviation 12g.
Assume weights are normally distributed.
Find the probability a randomly chosen bag weighs:
a) more than 1020g
b) between 990g and 1010g
(Use Z-scores; tables not required — explain your reasoning.)
---
Q10. A basketball player scores on 68% of their shots.
They take 20 shots in a match.
Find the probability they score at least 15 baskets.
(You may describe the method rather than compute the final decimal.)
---
19.7 Problem Set G — Real Mixed Problems
Q11 — The Two-Box Problem
Box A: 4 red, 6 blue
Box B: 7 red, 3 blue
A box is chosen at random, then a ball is drawn.
Find the probability the ball is red.
---
Q12 — The Airport Queue Problem
Security has two lanes:
• Lane 1: 70% of people
• Lane 2: 30% of people
Waiting times:
• Lane 1: mean 8 mins, SD 3 mins
• Lane 2: mean 4 mins, SD 2 mins
Find the mean overall waiting time of a random passenger.
---
Q13 — The Broken Graph Problem
A misleading graph shows a company’s profit rising sharply.
However, the Y-axis starts at £950,000 instead of £0.
Explain clearly why this exaggerates the company’s performance.
---
19.8 Mini Mastery Test
Try to answer these without looking back:
1. What is the difference between independent and dependent probability?
2. When should you use a tree diagram?
3. What does the expected value tell you?
4. Why might the mean NOT be the best measure of “typical”?
5. What does a Z-score of +1.5 represent?
---
19.9 Chapter Summary
• This chapter tests full-course knowledge
• Questions cover ratios, probability, distributions, expected value, averages
• Many problems require combining multiple ideas
• This is similar to the structure of real exam papers
• Being able to explain your reasoning is just as important as the final answer
You are now prepared for the final chapter — the Mastery Test.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
You’ve reached the chapter where we test EVERYTHING you’ve learned so far.
These questions are designed to feel like:
• GCSE Higher
• real-world tests
• logic puzzles
• mixed-topic challenges
Each problem requires you to think, combine ideas, and apply proper reasoning.
Work through them carefully — and avoid the temptation to guess.
---
19.1 Problem Set A — Ratios, Fractions & Basic Probability
Q1. A bag contains red, blue, and green counters in the ratio
3 : 5 : 2.
There are 50 counters in total.
Find how many of each colour there are.
---
Q2. A coin is biased.
It lands on heads with probability 0.62.
What is the probability of landing tails?
---
Q3. A spinner has 8 equal sections numbered 1–8.
A number is chosen at random.
Find the probability the number is:
• a multiple of 2
• greater than 5
• both (even AND > 5)
---
19.2 Problem Set B — Independent & Dependent Probability
Q4. A box contains:
• 6 black marbles
• 4 white marbles
Two marbles are drawn without replacement.
Find the probability that:
1. both are black
2. one black and one white (in any order)
---
Q5. A train is late 30% of the time.
If trains are independent day to day:
Find the probability the train is late exactly 2 times in a week (7 days).
(You may use the binomial formula or a clear reasoning method.)
---
19.3 Problem Set C — Conditional Probability
Q6. In a college:
• 40% study science
• 60% study arts
• 25% of science students play a sport
• 10% of arts students play a sport
A student is chosen at random.
They are found to play a sport.
Find the probability that they study science.
(You should use a tree diagram or conditional probability formula.)
---
19.4 Problem Set D — Tables, Frequency & Averages
Q7. A group of students were asked how many hours they revised last week.
Their results are shown:
Hours revised (h): 0, 1, 2, 3, 4
Frequency: 6, 8, 10, 5, 1
Calculate:
1. the mean
2. the median
3. the mode
4. the range
---
19.5 Problem Set E — Expected Value
Q8. A game costs £3 to play.
You draw a card at random:
• Win £10 (probability 0.15)
• Win £3 (probability 0.25)
• Win £1 (probability 0.40)
• Win £0 (probability 0.20)
Find the expected value, then determine whether the game is fair.
---
19.6 Problem Set F — Distributions (Normal & Binomial)
Q9. A machine fills bags of flour with mean 1000g and standard deviation 12g.
Assume weights are normally distributed.
Find the probability a randomly chosen bag weighs:
a) more than 1020g
b) between 990g and 1010g
(Use Z-scores; tables not required — explain your reasoning.)
---
Q10. A basketball player scores on 68% of their shots.
They take 20 shots in a match.
Find the probability they score at least 15 baskets.
(You may describe the method rather than compute the final decimal.)
---
19.7 Problem Set G — Real Mixed Problems
Q11 — The Two-Box Problem
Box A: 4 red, 6 blue
Box B: 7 red, 3 blue
A box is chosen at random, then a ball is drawn.
Find the probability the ball is red.
---
Q12 — The Airport Queue Problem
Security has two lanes:
• Lane 1: 70% of people
• Lane 2: 30% of people
Waiting times:
• Lane 1: mean 8 mins, SD 3 mins
• Lane 2: mean 4 mins, SD 2 mins
Find the mean overall waiting time of a random passenger.
---
Q13 — The Broken Graph Problem
A misleading graph shows a company’s profit rising sharply.
However, the Y-axis starts at £950,000 instead of £0.
Explain clearly why this exaggerates the company’s performance.
---
19.8 Mini Mastery Test
Try to answer these without looking back:
1. What is the difference between independent and dependent probability?
2. When should you use a tree diagram?
3. What does the expected value tell you?
4. Why might the mean NOT be the best measure of “typical”?
5. What does a Z-score of +1.5 represent?
---
19.9 Chapter Summary
• This chapter tests full-course knowledge
• Questions cover ratios, probability, distributions, expected value, averages
• Many problems require combining multiple ideas
• This is similar to the structure of real exam papers
• Being able to explain your reasoning is just as important as the final answer
You are now prepared for the final chapter — the Mastery Test.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive