Complement Rule
Probability that an event does not occur.
Formula
Variables
Example
P(not 6) = 1 - 1/6 = 5/6
How it works
An event and its complement are two outcomes that cover all possibilities: the event happens or it does not. Since their probabilities must add up to 1, the probability of the complement is found by subtracting the event’s probability from 1. This is useful whenever the “not happening” case is easier to calculate or is what the question asks for.
When to use it
Use this formula when you need the probability that something does not happen, such as not drawing a red card, not passing a test, or not getting a specific result. It is especially helpful when the probability of the event itself is already known or easier to find than counting the complement directly.
Worked example, step by step
A bag contains 12 marbles: 5 blue, 4 green, and 3 yellow. Find the probability of not drawing a blue marble. P(blue) = 5/12 P(not blue) = 1 - P(blue) P(not blue) = 1 - 5/12 P(not blue) = 7/12 Final answer: 7/12
Common mistakes to avoid
- Subtracting the event probability from the wrong total instead of from 1
- Forgetting that the complement includes every outcome that is not the event, not just one alternative outcome
- Using the formula when the event probability has not been found correctly first
Did You Know?
Often easier to compute the complement than the event directly.
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