Design and assess a fair random decision
Give competing outcomes equal winning probabilities · Assess fairness from the random mechanism · Explain assumptions and a tie or retry rule
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Counting events in a sample space
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Give competing outcomes equal winning probabilities
- Assess fairness from the random mechanism
- Explain assumptions and a tie or retry rule
Fairness needs the mechanism
For a choice between two students, equal winning probabilities make the random decision fair. A fair die assigning faces 1,2,3 to one student and 4,5,6 to the other gives each 1/2. Assigning just one face to the first student and the other five to the second is not fair.
A spinner is weighted by sector angles under an ideal uniform-angle model, not by the number of colour names. Equal colour labels can occupy unequal angles. Real bias or poor randomisation can invalidate the ideal model.
Handle ties before drawing
Two independent fair coins can decide: HT gives one student, TH gives the other, while HH and TT trigger a retry. The two winning patterns have equal probability 1/4 on each round. Repeated fresh rounds preserve symmetry; do not award every tie to one student. For three choices, a fair die can give two faces to each choice.
Sources for this lesson
Pause and explain
Does giving each of two students three faces of a fair die make the choice fair?
Worked example
Assess a fair die rule: faces 1–4 select Linh, faces 5–6 select Arun.
Show the worked solution
- Linh has four of six equally likely faces; Arun has two.
- Compare 2/3 with 1/3.
Answer The rule is unfair; assign three faces to each for equal probabilities.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A fair die selects Red on 1 or 2 and Blue otherwise. Find both probabilities and assess fairness.
Give me a hint
Count the face groups.
Compare my reasoning
- Red has 2/6=1/3.
- Blue has 4/6=2/3, so the probabilities differ.
Red 1/3; Blue 2/3; unfair.
Look for these in your work
- Use all six faces
- Compare winning probabilities
An ideal spinner has a 90° amber sector and a 270° purple sector. Is choosing by colour fair between two teams? Find both probabilities.
Give me a hint
Use sector angle divided by 360°.
Compare my reasoning
- Amber has 90/360=1/4.
- Purple has 270/360=3/4.
Amber 1/4; purple 3/4; unfair under the uniform-angle model.
Look for these in your work
- Weight sectors by angle
- State the random-angle assumption
Use one fair die to choose equally among three study activities. Give a rule and explain why it is fair.
Give me a hint
Assign equal numbers of faces.
Compare my reasoning
- Give 1,2 to activity A; 3,4 to B; 5,6 to C.
- Each activity has probability 2/6=1/3.
One valid rule is A on 1,2; B on 3,4; C on 5,6, giving 1/3 each.
Look for these in your work
- Cover every face exactly once
- Give all activities equal probability
Common mistakes
- Equal labels need not have equal probability.
- An asymmetric tie rule can introduce an advantage.
What can you explain now?
Does giving each of two students three faces of a fair die make the choice fair?
Compare with the explanation
Yes; each receives probability 3/6=1/2.
The outcomes assigned to the students have equal total probability.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does giving each of two students three faces of a fair die make the choice fair?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.