Probability scales and complementary events
Calculate equally likely outcomes and use the event that something does not happen.
Core and Extended.
Before you begin
- Simplify fractions and percentages.
- Count outcomes systematically.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which value is a possible probability?
What you will learn
- Calculate probability from an equally likely outcome set.
- Use complementary probabilities adding to one.
Probability measures how likely an event is
Probability is between zero and one inclusive: zero means impossible and one means certain. For equally likely outcomes, divide the number satisfying the event by the total number. On a fair six-sided die, probability of an even number is 3/6 = 1/2 because 2, 4 and 6 satisfy the event.
The equally likely condition matters. A loaded die still has six possible faces, but counting three even faces does not justify probability one half. Information can also come from a table, graph or Venn diagram; count the relevant group within the correct total.
An event and its complement exhaust the possibilities
The complement means the event does not occur. Their probabilities add to one, so if rain has probability 0.35, no rain has probability 0.65. Extended notation may write P(A) and P(A′); Core can state the events in words without requiring that notation.
If a bag has four red and six blue counters and one is selected randomly, not red means blue because these are the only colours. If there are additional colours, not red includes all of them. Define the event and outcome set before computing its complement.
Pause and explain
P(rain) = 0.35. Find P(no rain).
Worked example
A fair die is rolled. Find the probability of a number greater than four and of its complement.
Show the worked solution
- The event contains faces five and six, two out of six.
- Its probability is 1/3.
- The complement has probability 1−1/3 = 2/3, corresponding to faces one to four.
Answer Greater than four: 1/3; not greater than four: 2/3.
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 die is rolled. Find the probability of an even face.
Give me a hint
List the favourable faces.
Compare my reasoning
- The even faces are two, four and six.
- There are three favourable outcomes among six equally likely faces.
- Probability is 3/6 = 1/2.
1/2.
Look for these in your work
- Counted the event accurately.
- Used equally likely outcomes.
A bag has 5 red, 3 blue and 2 green counters. Find the probability of not red.
Give me a hint
Include every non-red colour.
Compare my reasoning
- The total number is ten.
- There are five non-red counters: three blue and two green.
- Probability is 5/10 = 1/2.
1/2.
Look for these in your work
- Used the complete total.
- Included both complement colours.
A spinner has four sectors of unequal sizes. Is each colour's probability 1/4 just because four colours are shown? Explain.
Give me a hint
Equal outcome names do not guarantee equal chance.
Compare my reasoning
- Unequal sector angles give different chances under an ideal fair spin.
- The probability of a colour depends on its fraction of the full turn.
- Counting four colours alone does not justify one quarter each.
No; use sector-angle proportions for an ideal spinner.
Look for these in your work
- Checked the equal-chance assumption.
- Identified the required information.
Common mistakes
- Assuming named outcomes are equally likely.
- Using only one of several complement categories.
- Reporting a probability outside zero to one.
What can you explain now?
Can an event and its complement happen together?
Compare with the explanation
No.
Exactly one of 'the event occurs' and 'the event does not occur' happens on a trial.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, define an event and its complement and check that the probabilities add to one.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources