C
CraftExam
medium difficulty5 min

Complement of an event

The complement of an event is the event that the original event does not happen.

In one line: Complement means the rest, so subtract from 1.

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Understand it

If the probability of an event is known, its complement helps us find the remaining chance. The two probabilities always add to 1. This is very useful when the direct counting is longer than counting the opposite case.

Example

If the probability of getting a head on a fair coin is 1/2, then the probability of not getting a head is also 1/2.

Formula and rule

P(not E) = 1 - P(E).

The event and its complement together cover all possibilities.

See it worked out

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

Worked example

  1. 1If P(not getting a 6) on a die = 1 - 1/6 = 5/6.
  2. 2It is easier to count the single face 6 and subtract from 1.

Method you can reuse

  1. 1Identify the event.
  2. 2Find or use its probability.
  3. 3Subtract it from 1.
  4. 4Write the complement clearly in words.

Visual tip

A two-part bar or a Venn-style split can show the event and its complement adding to 1.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If the probability of rain today is 3/10, what is the probability of no rain today?

Think first, then show answer
The probability of no rain today is 1 - 3/10 = 7/10, because an event and its complement add up to 1.

Common wrong answer

A common wrong answer is 3/10 for no rain because the student repeats the given probability instead of subtracting it from 1.

Don't mix this up

Students often subtract from the wrong number or confuse the complement with a completely different event.

Exam tip

Use complement when the opposite event is easier to count than the original event.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students are asked to find the probability of an event not happening, or to use 1 minus the given probability.
  1. 1

    Enough for 1 mark

    The complement of an event is the event that the original event does not happen.

  2. 2

    Add for 2 marks

    Example: If the probability of getting a head on a fair coin is 1/2, then the probability of not getting a head is also 1/2.

  3. 3

    Add for 3 marks

    If the probability of an event is known, its complement helps us find the remaining chance.

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