Complement of an Event
The complement of an event means all outcomes where the event does not happen. If an event is written as (A), its complement is often written as (A^c) or (A').
How to find it
- Identify the event clearly. Ask: what outcomes are included in the event?
- List the opposite outcomes. These are the outcomes that are not in the event.
- Write the complement using the same sample space, but without the event outcomes.
- Simplify the answer if possible. If the problem gives a probability, you can use:
[P(A^c)=1-P(A).]
Useful idea
The event and its complement together make the whole sample space. That means their probabilities add to 1:
[P(A)+P(A^c)=1.]
Check your work
- Make sure no outcome is counted in both the event and its complement.
- Make sure every possible outcome is included in exactly one of the two.
- If you used probabilities, verify that the result is between 0 and 1.
This method works for both outcome lists and probability calculations. Read the event carefully, find what is excluded, and simplify your final result.