Year 8 - Complementary events and probability - Free practice

Year 8 Complementary Events and Probability Worksheets & Practice

Understand why an event and its complement always add to 1, with worked solutions every time, free.

Quick answer

What are complementary events in probability?

Complementary events are two outcomes that together cover every possibility, so their probabilities always add to 1. Use P(not A) = 1 − P(A) to find a missing probability. For example, if the probability of rain is 0.3, the probability of no rain is 1 − 0.3 = 0.7.

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Worked examples

Year 8 complementary events and probability: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Finding the complement of an event
A bag contains 20 marbles. The probability of picking a red marble at random is $\dfrac{3}{20}$ . Find the probability of not picking a red marble.
Show worked solution
Key idea: Complementary events cover all possible outcomes, so $P(\text{not A}) = 1 - P(A)$ .
1. Identify the given probability: $P(\text{red}) = \dfrac{3}{20}$
2. Apply the complement rule:
  $P(\text{not red}) = 1 - \dfrac{3}{20}$
3. Subtract:
  $P(\text{not red}) = \dfrac{20}{20} - \dfrac{3}{20} = \dfrac{17}{20}$
Answer: $\dfrac{17}{20}$
Example 2 · Finding a probability using the complement rule
The probabilities of spinning a spinner and landing on each colour are shown below.

Colour Red Blue Green Yellow
Probability $0.35$ $0.2$ $0.3$ ?

Find the probability of landing on yellow.
Show worked solution
Key idea: All probabilities in a sample space must sum to 1, so the missing probability equals $1$ minus the sum of the rest.
1. Add the known probabilities:
  $0.35 + 0.2 + 0.3 = 0.85$
2. Apply the complement rule:
  $P(\text{yellow}) = 1 - 0.85$
3. Subtract:
  $P(\text{yellow}) = 0.15$
Answer: $0.15$
Example 3 · Applying the complement rule in a real-world context
A school raffle sells 250 tickets. Mia buys 8 tickets. All tickets are equally likely to win. Find the probability that Mia does not win, giving your answer as a fraction, a decimal and a percentage.
Show worked solution
Key idea: It is easier to find $P(\text{win})$ first, then use the complement to find $P(\text{not win})$ .
1. Find $P(\text{Mia wins})$ :
  $P(\text{win}) = \dfrac{8}{250} = \dfrac{4}{125}$
2. Apply the complement rule:
  $P(\text{not win}) = 1 - \dfrac{4}{125} = \dfrac{125}{125} - \dfrac{4}{125} = \dfrac{121}{125}$
3. Convert to a decimal:
  $\dfrac{121}{125} = 0.968$
4. Convert to a percentage:
  $0.968 \times 100 = 96.8\%$
Answer: $\dfrac{121}{125}$ , $0.968$ , $96.8\%$

Colour	Red	Blue	Green	Yellow
Probability	$0.35$	$0.2$	$0.3$	?

Common mistakes

Where students usually lose marks

Subtracting from the wrong starting value

Student writes $P(\text{not red}) = 20 - 3 = 17$ instead of $\dfrac{17}{20}$

Always subtract from 1, not from the total number of outcomes. Probability is always a value between 0 and 1, so $P(\text{not A}) = 1 - P(A)$ .

Forgetting that all probabilities must sum to 1

Student finds a missing probability by guessing or leaving it blank when a table of probabilities is given.

Add all the known probabilities first, then subtract their total from 1. This always gives the missing probability, because every sample space must have probabilities that sum to exactly 1.

Confusing the event with its complement

For the event 'not rolling a 6', the student writes $P(6) = \dfrac{5}{6}$ instead of $P(\text{not } 6) = \dfrac{5}{6}$ .

Read the question carefully and check exactly which event you need. The complement of rolling a 6 is rolling any other number (1, 2, 3, 4 or 5), which has 5 favourable outcomes out of 6.

Converting incorrectly between fractions, decimals and percentages

Student writes $\dfrac{17}{20} = 17.20\%$ instead of $85\%$ .

To convert a fraction to a percentage, divide the numerator by the denominator to get a decimal, then multiply by 100. For example, $\dfrac{17}{20} = 0.85 = 85\%$ .

Try one yourself

Your turn

A bag contains only red and blue marbles. The probability of picking a red marble is $\dfrac{3}{8}$ . What is the probability of picking a blue marble? Give your answer as a fraction.

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About this worksheet

Why we built it

Complementary events are a key idea in Year 8 probability: if you know the chance of something happening, you automatically know the chance of it not happening, because the two probabilities always add to 1. These worksheets help students move from that core rule to applying it confidently in real contexts, with worked solutions for every question so they can check their reasoning step by step.

What's covered

Sub-skills your student will practise

✓Identifying an event and its complement
✓Using P(not A) = 1 minus P(A) to find missing probabilities
✓Expressing probabilities as fractions, decimals and percentages
✓Applying the complement rule to worded real-world problems
✓Checking that all probabilities in a sample space sum to 1
✓Recognising when using the complement is the most efficient approach

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Frequently asked

Questions parents ask about complementary events and probability

What is a complementary event in Year 8 maths?

A complementary event is everything that can happen apart from the event you are looking at. For example, if the probability of rolling a 6 is 1/6, the complementary event is not rolling a 6, which has a probability of 5/6. The two probabilities always add to 1.

How do students use the complement rule to solve problems?

Students rearrange the rule P(not A) = 1 minus P(A). If a problem gives the probability of an event happening, they subtract it from 1 to find the probability of it not happening. This is often quicker than listing every other outcome.

Why do complementary events add to 1 and not 100?

Probability is measured on a scale from 0 to 1, where 1 means certain. When you write probability as a percentage it adds to 100%, but as a decimal or fraction it adds to 1. Both are correct, just different ways of expressing the same idea.

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