Compare observed results with a probability model
State assumptions for an experimental model · Use a specified pattern or simulation as evidence · Separate questioning a model from disproving it
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
- Relative frequency
- Simple multiplication of successive branch probabilities
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
- State assumptions for an experimental model
- Use a specified pattern or simulation as evidence
- Separate questioning a model from disproving it
Possible does not mean typical
Under independent fair coin tosses, a specified sequence of five tails has probability (1/2)⁵=1/32. It is possible and is not proof of bias. “Five tails in these next five tosses” is different from “some run of five tails anywhere in a long record”: the latter has many opportunities. Decide which observation was specified before calculating.
A result unusually far from model predictions can justify checking the coin, randomisation and assumptions. It cannot by itself logically disprove a model that gives that result positive probability. Formal significance testing is not required for this lesson.
Use simulation honestly
A fair independent coin can be simulated by independent uniformly random digits, assigning 0–4 to heads and 5–9 to tails. Repeat whole experiments of the same size to compare with a recorded result. Repeating a fixed HTHT pattern is not the same random process. Simulated results illustrate variability; they do not validate an incorrect mapping or a biased source of digits.
Sources for this lesson
Pause and explain
Does a positive-probability unusual result logically disprove the model?
Worked example
A claimed independent fair coin gives tails in its next five specified tosses. Does that make the model impossible?
Show the worked solution
- The specified sequence probability is 1/32.
- It is non-zero; further checks and observations can investigate the model.
Answer No; this sequence is possible under the model, although it can prompt investigation.
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.
Under independent fair coin tosses, find the probability of tails on the next five specified tosses. Is it zero?
Give me a hint
Multiply five factors of 1/2.
Compare my reasoning
- The path probability is (1/2)⁵.
- It equals 1/32=0.03125.
1/32 or 3.125%; not zero.
Look for these in your work
- Specify the five tosses
- State independence and fairness
Uniform independent digits 0–9 simulate a claimed fair coin. A student maps 0–3 to heads and 4–9 to tails. Correct the mapping and explain the bias.
Give me a hint
Count the digit categories.
Compare my reasoning
- The proposed head probability is 4/10, not 1/2.
- Use five digits for each outcome, such as 0–4 and 5–9.
The proposed simulation is biased: heads 0.4, tails 0.6; split the digits five and five.
Look for these in your work
- Use equally likely input digits
- Match the claimed model
Of 1000 simulated 20-toss experiments, 942 have 6–14 heads and 58 are outside that range. An actual experiment gives 3 heads. What can this comparison support, and what can it not prove?
Give me a hint
Compare the observation with the stated simulation range.
Compare my reasoning
- Three heads is outside the common simulated range.
- The 58 outside results include different extremes; they are not all exactly 3 heads. The comparison supports investigation, not logical disproof or an exact probability for 3.
It supports questioning and checking the model; it does not prove bias or give P(exactly 3 heads)=58/1000.
Look for these in your work
- Use the same experiment size
- Distinguish a range count from an exact-result count
Common mistakes
- Define the observation and number of opportunities.
- Check simulation assumptions before trusting its output.
What can you explain now?
Does a positive-probability unusual result logically disprove the model?
Compare with the explanation
No; it can be evidence to question the model without making it impossible.
A model can permit outcomes that rarely occur.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does a positive-probability unusual result logically disprove the model?
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.