Use weighted paths to interpret a later outcome
Combine source probabilities with outcome probabilities · Find the total probability of a later event · Compare source shares within that event
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Tree paths
- Restricted 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
- Combine source probabilities with outcome probabilities
- Find the total probability of a later event
- Compare source shares within that event
Different branch rates need source weights
An illustrative production model supplies 60% of items from A and 40% from B. Rejection rates are 5% for A and 10% for B. The complete rejected paths have probabilities 0.6×0.05=0.03 and 0.4×0.10=0.04, so overall rejection is 0.07. Averaging 5% and 10% equally to get 7.5% ignores the different source shares.
This is an invented teaching model, not evidence about any real supplier. The branches after each source sum to 1. Complete path probabilities are measured relative to all items.
Given rejection, change the reference group
Among rejected items, B accounts for 0.04 of the total rejection probability 0.07. The conditional share is 4/7. It is not 10%: that value is the rejection chance given B, the reverse conditioning direction. A higher rejection rate does not by itself guarantee a larger share of rejected items, because the source size also matters.
Sources for this lesson
Pause and explain
Is the chance of source B given rejection equal to its 10% rejection rate?
Worked example
For the illustrated model, find P(from B given rejected).
Show the worked solution
- B-and-rejected probability=0.04; total rejected=0.07.
- Divide the B path by the total rejected probability.
Answer 4/7.
Try a topic test
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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 the illustrated A/B model, find the overall probability an item is rejected.
Give me a hint
Add the two rejected paths.
Compare my reasoning
- A-and-rejected=0.03.
- B-and-rejected=0.04; total=0.07.
0.07 or 7%.
Look for these in your work
- Weight by source share
- Add disjoint rejected paths
Use the same model to find P(source A given accepted).
Give me a hint
Find A-and-accepted and total accepted.
Compare my reasoning
- A-and-accepted=0.6×0.95=0.57.
- Accepted probability=1−0.07=0.93; ratio=0.57/0.93.
19/31.
Look for these in your work
- Use the accepted reference group
- Normalise the selected path
Another model has A supplying 90% with a 2% rejection rate, and B supplying 10% with a 10% rate. Which source supplies more rejected items? Find the source-B share among rejected items.
Give me a hint
Compare weighted paths, not just branch rates.
Compare my reasoning
- A-and-rejected=0.9×0.02=0.018; B-and-rejected=0.1×0.10=0.010.
- A contributes more; B share=0.010/0.028=5/14.
A supplies more rejected items; B share is 5/14.
Look for these in your work
- Account for unequal source sizes
- Restrict to the rejected population
Common mistakes
- Weight each rate by its source share.
- Reversing the word given reverses the reference group.
What can you explain now?
Is the chance of source B given rejection equal to its 10% rejection rate?
Compare with the explanation
No; the source share among rejected items is 4/7.
The 10% rate conditions on source B, while the question conditions on rejection.
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Make it stick
Is the chance of source B given rejection equal to its 10% rejection rate?
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.