Calculate single-event probabilities and complements
Count equally likely outcomes, express probability from 0 to 1, and use complements and sets to reason about events.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the probability of a certain event?
What you will learn
- Interpret probability on the 0–1 scale
- Calculate and express a single-event probability
- Find the probability of an event not occurring using a complement
Count favourable outcomes in the whole sample space
For equally likely outcomes, P(A) = number of outcomes in A ÷ total outcomes. Probability lies from 0 to 1; 0 is impossible and 1 is certain.
Give a result as a fraction, decimal or percentage. Check that the denominator describes all equally likely possibilities, not only the favourable ones.
Use the complement for ‘not’
The complement A′ means the event A does not happen, and P(A′) = 1 − P(A). Extended candidates use probability notation such as P(A), P(A′), P(A ∩ B) and P(A ∪ B). Core questions do not require probability notation.
A two-set Venn diagram separates ‘A only’, ‘B only’, ‘both’ and ‘neither’. Use the universal set when finding a complement; in Core, Venn diagrams are limited to two sets.
Pause and explain
A student chooses one of 8 equally likely cards and 3 are stars. What is P(star)?
Worked example
A bag contains 5 red, 3 blue and 2 green counters. Find the probability that a randomly chosen counter is not blue.
Show the worked solution
- There are 10 counters in total and 3 are blue.
- Not blue means red or green: 5 + 2 = 7 counters.
- Probability = 7/10 = 0.7.
Answer 7/10 or 0.7.
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 six-sided die is rolled. Find the probability of an even number and of not rolling an even number.
Give me a hint
List the even results and use the complement for the second part.
Compare my reasoning
- Even results are 2, 4 and 6, so P(even) = 3/6 = 1/2.
- The complement is 1 − 1/2.
- P(not even) = 1/2.
P(even) = 1/2; P(not even) = 1/2.
Look for these in your work
- Favourable outcomes listed
- Fraction simplified
- Complement checked
In a group of 30 learners, 18 study French, 12 study Korean and 5 study both. How many study neither, and what is the probability that a randomly selected learner studies at least one?
Give me a hint
Add the two set counts, subtract the overlap once, then compare with the universal total.
Compare my reasoning
- At least one = 18 + 12 − 5 = 25.
- Neither = 30 − 25 = 5.
- Probability of at least one = 25/30 = 5/6.
5 study neither; probability 5/6 study at least one.
Look for these in your work
- Overlap counted once
- Complement within all 30
- Probability simplified
A quality check reports that 0.92 of items pass. Find the probability an item fails, then give the expected number of failures in a random sample of 50 if that rate continues.
Give me a hint
Use a complement first; multiply the failure probability by the sample size for an expectation.
Compare my reasoning
- P(fail) = 1 − 0.92 = 0.08.
- Expected failures = 50 × 0.08.
- The expected count is 4.
Failure probability 0.08; expected failures 4.
Look for these in your work
- Complement used
- Expected value model set up
- Count interpreted
Common mistakes
- Using the favourable count as the denominator
- Adding probabilities of overlapping events without removing the intersection
- Treating an event and its complement as independent
- Forgetting the universal set when listing the complement
What can you explain now?
If P(A) = 0.35, find P(A′).
Compare with the explanation
0.65
The two complementary probabilities sum to 1.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does a complement help when a question asks for an event not occurring?
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.