Formula sheet
Complement of an event
P(A') = 1 − P(A) is given on the SPM formula sheet. You must know when it saves time, when counting 'not A' is easier than 'A', for example 'at least one' problems, and that A and A' must together cover the whole sample space with no overlap.
What the symbols mean
- P(A') the probability of the complement of A (A not happening)
- P(A) the probability of event A
- A' the complement of A: all outcomes in the sample space that are not in A
Given in the exam, or memorise?
P(A') = 1 − P(A) is given on the SPM formula sheet. You must know when it saves time, when counting 'not A' is easier than 'A', for example 'at least one' problems, and that A and A' must together cover the whole sample space with no overlap.
Why it works
Every outcome either is in A or is not, so the two probabilities must add up to a whole, which is 1.
- The event A and its complement A' together include every possible outcome and share none.
- So their probabilities cover the whole sample space: P(A) + P(A') = 1.
- Rearranging gives P(A') = 1 − P(A).
- This lets you find the harder probability by subtracting the easier one from 1.
Worked example 1
The probability that it rains tomorrow is P(A) = 0.35. Find the probability that it does not rain.
- Identify the event: A = 'it rains', with P(A) = 0.35.
- 'Does not rain' is the complement A', and rain/no-rain cover all outcomes.
- Apply the formula: P(A') = 1 − P(A) = 1 − 0.35.
Worked example 2
In a class of 30 students, 18 wear glasses. A student is chosen at random.
Find the probability that the student does not wear glasses.
- Let A = 'wears glasses': P(A) = n(A)/n(S) = 18/30 = 3/5.
- 'Does not wear glasses' is the complement A'.
- Apply the formula: P(A') = 1 − P(A) = 1 − 3/5.
- Subtract: 1 − 3/5 = 2/5.
Where students go wrong
- Counting 'not A' the long way when 1 − P(A) is far quicker, the whole point of the formula is to avoid that.
- Forgetting the '1 −' and just reporting P(A) as the answer.
- Using the complement rule when A and the other event are not a true complementary pair (they must cover the whole sample space with no overlap).
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What does the complement of an event mean?
The complement A' is 'A does not happen' every outcome in the sample space that is not in A. For rolling a die, if A is 'getting a 6', then A' is 'not getting a 6', that is {1,2,3,4,5}.
Together A and A' account for all possible outcomes, so their probabilities add to 1.
When should I use 1 − P(A) instead of counting directly?
Use it whenever 'not A' is easier to find than A. The classic case is 'at least one', where listing every success is tedious but the single 'none' case is quick: P(at least one) = 1 − P(none).
If A itself is simple, just compute P(A) directly instead.
Does P(A) + P(A') always equal 1?
Yes, as long as A' is the true complement of A. Since every outcome is either in A or in A' and never both, their probabilities must fill the whole sample space, giving P(A) + P(A') = 1.
This identity is exactly what the formula P(A') = 1 − P(A) rearranges.