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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

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

Memory hook

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

Examples and method

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 to apply

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

Diagram support

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

How CBSE asks it

Questions may ask for not getting, not selected, at least one failure, or the probability left after an event.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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

Study the events and their complements diagram carefully

Use the labelled diagram to keep events and their complements clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of events and their complements in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on events and their complements.

Revision cue

Revise events and their complements through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

An event is a selected part of the sample space, and its complement is the event that it does not happen. P(E) + P(not E) = 1, so P(not E) = 1 - P(E). For a die, if E is getting an even number, then not E is getting an odd number.

3-mark answer

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. P(E) + P(not E) = 1, so P(not E) = 1 - P(E). If probability of a student being absent is 3/20, then probability of the student being present is 1 - 3/20 = 17/20. Questions may ask for not getting, not selected, at least one failure, or the probability left after an event. If E is getting more than 4 on a die, saying not E is {1, 2, 3} is wrong because 4 is also not more than 4.
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