Probability of something not happening
Complementary probabilities help you find the probability that something does not happen.
Since the probability of an event happening and not happening always adds up to 1 (or 100%), you can easily find the complement by subtracting the known probability from 1.
Practise now
Worksheet preview and key skills
Worksheet preview
This worksheet provides 12 self-marking practice questions on finding the probability that an event does not happen.
You will use the rule for probability complements, subtracting from 1 (or 100%) to find the missing probability.
What you’ll practise
- Finding complementary probabilities using decimals.
- Finding complementary probabilities using percentages.
- Finding the probability of a "not" event using fractions.
- Understanding complements for certain and impossible events.
Use the interactive worksheet below, or read the Topic guide for the method and worked example.
Calculate the probability that the event does not happen. Pay attention to the requested format for your answer.
Topic guide
In mathematics, the complement of an event is the probability that the event does not happen.
Because the sum of all possible mutually exclusive outcomes must equal 1, the probability of an event happening and the probability of it not happening will always add up to 1.
We can write this as a key rule:
P(not A) = 1 − P(A)
Worked example: Decimals
Question: The probability that it rains tomorrow is 0.3. What is the probability that it does not rain?
Method: Subtract the probability of it raining from 1.
1 − 0.3 = 0.7
Worked example: Fractions
Question: A bag contains 4 red counters, 3 blue counters and 5 green counters. What is the probability of choosing a counter that is not red?
Method 1: Subtracting from 1
First, find the total number of counters: 4 + 3 + 5 = 12.
The probability of choosing a red counter is 4/12 (or 1/3).
To find the probability of not choosing a red counter, subtract this from 1: 1 − 4/12 = 8/12 (which simplifies to 2/3).
Method 2: Counting the complement
The counters that are not red are the blue and green ones.
There are 3 + 5 = 8 counters that are not red.
So, the probability is 8/12 (or 2/3).
Common mistakes to avoid
- Forgetting to subtract from 1: Sometimes students work out the probability of the event happening, but forget the final step of subtracting from 1 to find the "not" probability.
- Mixing up formats: Make sure you subtract from 1 for decimals and fractions, but subtract from 100 for percentages!
- Forgetting the total in a raffle: If 500 tickets are sold and 5 are winning tickets, the number of losing tickets is 495. The probability of not winning is 495/500, not just 495.
Recap
To find the probability that an event does not happen, subtract the probability that it does happen from 1. Always check that your final answer is between 0 and 1 (or 0% and 100%).