Combine overlapping events without counting the overlap twice
Identify the common part of two events · Use the inclusive addition rule · Check event probabilities describe a possible model
Section Paper 2 · Historical Extended · non-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
- Or and and events
- Fractions
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
- Identify the common part of two events
- Use the inclusive addition rule
- Check event probabilities describe a possible model
Add, then remove one duplicate
P(A or B)=P(A)+P(B)−P(A and B). Adding the two event probabilities counts outcomes in both twice, so subtract the common part once. If events cannot occur together, their common part is zero and simple addition works. Do not assume there is no overlap merely because the events have different names.
For 40 students, 24 attend art, 18 music and 10 both. Art only is 14, music only 8, both 10 and neither 8. Either club has 32 students. The drawn region sizes are illustrative, not proportional to their counts.
Check consistency
An overlap cannot exceed either event probability. The union cannot exceed 1. These constraints can identify impossible data: P(A)=0.7, P(B)=0.6, overlap 0.1 would give union 1.2. The overlap must be at least 0.3 in that example, as well as at most 0.6.
Sources for this lesson
Pause and explain
When events overlap, is P(A or B)=P(A)+P(B) without adjustment?
Worked example
Find the probability of art or music for a uniform choice from the illustrated 40 students.
Show the worked solution
- Use (24+18−10)/40.
- The 32 distinct students give 32/40.
Answer 4/5.
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.
In 40 students, art has 24, music 18 and both 10. Find P(neither) for a uniform student choice.
Give me a hint
Find the union first.
Compare my reasoning
- Union count=24+18−10=32.
- Neither count=40−32=8.
1/5.
Look for these in your work
- Correct the duplicate count
- Complement the union
A model has P(A)=0.45, P(B)=0.35 and P(A and B)=0.15. Find P(A or B) and P(A only).
Give me a hint
Subtract the overlap in each appropriate calculation.
Compare my reasoning
- Union=0.45+0.35−0.15=0.65.
- A only=0.45−0.15=0.30.
Union 0.65; A only 0.30.
Look for these in your work
- Distinguish union from one-only
- Use the stated overlap
Can P(A)=0.7, P(B)=0.6 and P(A and B)=0.1 form a probability model? Explain the constraint that fails.
Give me a hint
Calculate the implied union.
Compare my reasoning
- The union would be 0.7+0.6−0.1=1.2.
- A probability cannot exceed 1; overlap must be at least 0.3 here.
No; the implied union is 1.2, so the data are inconsistent.
Look for these in your work
- Check probabilities as a complete model
- Do not accept an impossible union
Common mistakes
- Or includes both.
- Subtract the overlap once after adding the two event totals.
What can you explain now?
When events overlap, is P(A or B)=P(A)+P(B) without adjustment?
Compare with the explanation
No; subtract P(A and B) once.
The simple sum counts the common outcomes twice.
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Make it stick
When events overlap, is P(A or B)=P(A)+P(B) without adjustment?
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.