medium difficulty4 min

Events and their Complements

An event is a selected part of the sample space, and its complement is the event that it does not happen.

In one line: Complement means everything outside the event, not just the opposite word that first comes to mind.

Understand it

If E is an event, then not E includes all outcomes outside E. Since an event and its complement together cover the full sample space, their probabilities add to 1.

Example

For a die, if E is getting an even number, then not E is getting an odd number.

Formula and rule

P(E) + P(not E) = 1, so P(not E) = 1 - P(E).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

If probability of a student being absent is 3/20, then probability of the student being present is 1 - 3/20 = 17/20.

Method you can reuse

Write the event clearly, describe what does not satisfy it, then subtract its probability from 1 if needed.

Visual tip

A simple rectangle for sample space split into E and not E can show the complement idea.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If P(E) = 0.35, what is P(not E)?

Think first, then show answer

P(not E) is 0.65 because the probability of an event and its complement always add up to 1.

Common wrong answer

If E is getting more than 4 on a die, saying not E is {1, 2, 3}.

Why it loses marks: 4 is also not more than 4.

Don't mix this up

Students sometimes think the complement of greater than 4 is less than 4, forgetting the outcome equal to 4.

Exam tip

When finding a complement, include every outcome that does not satisfy the event, including boundary values.