Multiply marginal probabilities only when events are independent
State what independence means · Use the product rule for independent events · Distinguish marginal and history-dependent probabilities
Section Paper 4 · Historical Extended · calculator
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
- Addition rule
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 what independence means
- Use the product rule for independent events
- Distinguish marginal and history-dependent probabilities
The condition matters
Events A and B are independent when P(A and B)=P(A)P(B). If knowing one has occurred changes the chance of the other, do not multiply their unchanged marginal probabilities. A fresh uniformly mixed replaced draw is independent of the first draw in the stated model; an unreplaced draw generally is not.
For independent events with probabilities 0.3 and 0.4, both has probability 0.12. Either has probability 0.3+0.4−0.12=0.58. Exactly one has probability 0.3×0.6+0.7×0.4=0.46.
Different probabilities can still be independent
Independence does not mean the event probabilities are equal. Nor does a physical separation by itself prove independence: weather, shared equipment or a common cause can connect processes. In a problem, use an explicit independence assumption or evidence consistent with the product condition.
Sources for this lesson
Pause and explain
Must two independent events have the same probability?
Worked example
Independent events have P(A)=0.3 and P(B)=0.4. Find P(both).
Show the worked solution
- The independence condition permits multiplying marginals.
- 0.3×0.4=0.12.
Answer 0.12.
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.
Independent events have probabilities 1/4 and 2/5. Find P(both).
Give me a hint
Multiply the two marginals.
Compare my reasoning
- (1/4)(2/5)=2/20.
- Simplify to 1/10.
1/10.
Look for these in your work
- State independence
- Simplify the fraction
Independent events have probabilities 0.3 and 0.4. Find P(either) and P(exactly one).
Give me a hint
Find both, then distinguish inclusive or from exactly one.
Compare my reasoning
- Both=0.12; either=0.3+0.4−0.12=0.58.
- Exactly one=0.3×0.6+0.7×0.4=0.46.
Either 0.58; exactly one 0.46.
Look for these in your work
- Include both for either
- Exclude both for exactly one
Two red draws from 3 red and 2 blue without replacement each have marginal probability 3/5. Can you calculate the joint as (3/5)²? Find the correct joint.
Give me a hint
The second chance after red is 2/4.
Compare my reasoning
- The marginal probabilities being equal does not make the draws independent.
- Joint=(3/5)(2/4)=3/10, whereas the product of marginals is 9/25.
No; the joint is 3/10, not 9/25.
Look for these in your work
- Use the changed branch state
- Compare joint and marginal product
Common mistakes
- Multiply marginals only with the independence condition.
- Equal probabilities do not establish independence.
What can you explain now?
Must two independent events have the same probability?
Compare with the explanation
No; independence is the joint-product condition.
Different event probabilities can still multiply to the joint probability.
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Make it stick
Must two independent events have the same probability?
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.