Complementary Events
Learn about complementary events for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). An event’s complement is that event not happening — and since the two together cover every outcome, their probabilities add to 1.
You will use the rule \(P(\text{not } A) = 1 - P(A)\) to find the chance that something does not happen, working with fractions, decimals and percentages, and answer “not” and “at least one” questions quickly — such as the chance a flight is not delayed.
Theory
Complementary events are an event and its opposite, “not \(A\)”. This Year 12 Standard 1 (NSW) guide shows how the rule \(P(\text{not }A)=1-P(A)\) works with fractions, decimals and percentages, and how to solve “neither” and “at least one” questions quickly.
The complement of an event \(A\) is the event “\(A\) does not happen”, written “not \(A\)”. Every outcome is either in \(A\) or in not \(A\), never both.
Because an event and its complement cover every outcome with no overlap, their probabilities add to \(1\). So \(P(\text{not }A)=1-P(A)\), and also \(P(A)=1-P(\text{not }A)\).
Two events are complementary when together they fill the whole sample space with no overlap, e.g. “even” \(\{2,4,6\}\) and “odd” \(\{1,3,5\}\) on a die. Use the complement when “not” is quicker to find: “at least one” is the complement of “none”.
An event and its complement add to \(1\):
Rearranged, this is the complement rule:
With fractions, write \(1\) as a matching fraction: \(1-\dfrac{1}{6}=\dfrac{6}{6}-\dfrac{1}{6}=\dfrac{5}{6}\). With decimals subtract from \(1\) (\(1-0.3=0.7\)); with percentages subtract from \(100\%\) (\(100\%-8\%=92\%\)).
How to use the complement
- Name the event and its complement. For “at least one”, the complement is “none”.
- Find the probability that is easier to work out, as a fraction, decimal or percentage.
- Subtract from \(1\) (or from \(100\%\)), and check the two probabilities add back to \(1\).
Count the letters and the A’s, find \(P(\text{A})\), then take the complement.
| \(n(S)\) | \(=\) | \(10\) |
| \(n(\text{A})\) | \(=\) | \(4\) |
| \(P(\text{A})\) | \(=\) | \(\dfrac{4}{10}=\dfrac{2}{5}\) |
| \(P(\text{not A})\) | \(=\) | \(1-\dfrac{2}{5}\) |
| \(\) | \(=\) | \(\dfrac{5}{5}-\dfrac{2}{5}=\dfrac{3}{5}\) |
\(P(\text{not A})=\dfrac{3}{5}\).
(ii) On another line, \(P(\text{not late})=0.93\). Find \(P(\text{late})\).
The two probabilities add to \(1\), so subtract from \(1\) each time.
| \(\text{(i) }P(\text{not late})\) | \(=\) | \(1-0.12=0.88\) |
| \(\text{(ii) }P(\text{late})\) | \(=\) | \(1-P(\text{not late})\) |
| \(\) | \(=\) | \(1-0.93=0.07\) |
(i) \(0.88\) (ii) \(0.07\)
Percentages add to \(100\%\), so subtract from \(100\%\).
| \(\text{(i) }P(\text{not orange})\) | \(=\) | \(100\%-45\%=55\%\) |
| \(\text{(ii) }P(\text{apple or orange})\) | \(=\) | \(30\%+45\%=75\%\) |
| \(P(\text{neither})\) | \(=\) | \(100\%-75\%=25\%\) |
(i) \(55\%\) (ii) \(25\%\)
The complement of “at least one even” is “no even”, i.e. both dice odd. Each die has \(3\) odd faces, so \(3\times3=9\) of the \(36\) outcomes are odd–odd.
| \(n(\text{no even})\) | \(=\) | \(3\times3=9\) |
| \(P(\text{no even})\) | \(=\) | \(\dfrac{9}{36}=\dfrac{1}{4}\) |
| \(P(\text{at least one even})\) | \(=\) | \(1-\dfrac{1}{4}=\dfrac{3}{4}\) |
\(P(\text{at least one even})=\dfrac{3}{4}\).
Common pitfalls
Frequently asked questions
What is a complementary event?
The complement of an event A is the event that A does not happen, written not A. Together A and not A cover every outcome with no overlap, so their probabilities add to 1.
What is the formula for the complement of an event?
P(not A) = 1 minus P(A). With percentages it is 100 percent minus P(A). For example, a 0.12 chance of a late train gives a 0.88 chance it is not late.
How do you find the probability of at least one?
Find the probability of none, then subtract it from 1. For two dice, the chance of no even number is 9 over 36, which is 1 quarter, so the chance of at least one even number is 3 quarters.
How do you find the probability of neither A nor B?
Add P(A) and P(B) when they cannot both happen, then subtract from 1. If 30 percent are apples and 45 percent are oranges, then 25 percent are neither.
Are even and odd complementary events?
Yes. On a die, even is {2, 4, 6} and odd is {1, 3, 5}. Together they cover all six outcomes with no overlap, so P(even) + P(odd) = 1.
Do P(A) and P(not A) always add to 1?
Yes. An event and its complement include every outcome in the sample space exactly once, so their probabilities always add to 1, or 100 percent.