Complementary Events
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Theory
The complement of an event \(A\) is “not \(A\)”: every outcome in the sample space that is not in \(A\). An event and its complement together make up all the outcomes, so \(P(A) + P(\text{not } A) = 1\). This Year 7 maths page shows how to describe complements and use the complement rule.
The complement of an event \(A\) is “not \(A\)”. List the sample space, then take away the outcomes in \(A\). For one roll of a die, the complement of “an even number” \(\{2, 4, 6\}\) is “not an even number” \(\{1, 3, 5\}\). The complement of \(E\) is sometimes written \(\overline{E}\).
An event and its complement have no outcomes in common, and together they make up the whole sample space. They are complementary events, and their probabilities add to \(1\).
The complement rule: \(P(\text{not } A) = 1 - P(A)\). It works in every form: \(1 - \dfrac{4}{9} = \dfrac{9}{9} - \dfrac{4}{9} = \dfrac{5}{9}\); \(1 - 0.3 = 0.7\); \(100\% - 22\% = 78\%\).
For “neither \(A\) nor \(B\)”, first count “\(A\) or \(B\)” (each outcome once), then take it away from the total.
The probabilities of all the outcomes add to \(1\). Use this to find a missing probability.
The complement rule:
How to find the probability of a complement
- Find \(P(A)\), or read it from the question.
- Take it away from \(1\) (or from \(100\%\)).
- Check: the two probabilities add to \(1\).
How to find “neither”
- Count the outcomes in “\(A\) or \(B\)”, each once.
- Take them away from the total, then divide by the total.
(i) List the sample space.
(ii) Find \(P(4)\).
(iii) List the outcomes in the complement of “spinning a \(4\)”.
(iv) Find \(P(\text{not } 4)\).
| \(1 - \dfrac{1}{5}\) | \(=\) | \(\dfrac{5 - 1}{5} = \dfrac{4}{5}\) |
(i) \(\{1, 2, 3, 4, 5\}\)
(ii) \(\dfrac{1}{5}\) (\(1\) chance in \(5\))
(iii) \(\{1, 2, 3, 5\}\)
(iv) \(\dfrac{4}{5}\)
(i) Describe the complement of “a number greater than \(6\)”.
(ii) Find the probability of the event and of its complement.
| \(\dfrac{2}{8}\) | \(=\) | \(\dfrac{1}{4}\) |
| \(\dfrac{6}{8}\) | \(=\) | \(\dfrac{3}{4}\) |
| \(\dfrac{1}{4} + \dfrac{3}{4}\) | \(=\) | \(1\) |
(i) The event is \(\{7, 8\}\); its complement is “\(6\) or less”: \(\{1, 2, 3, 4, 5, 6\}\).
(ii) \(\dfrac{1}{4}\) and \(\dfrac{3}{4}\)
(ii) There is a \(40\%\) chance of a storm today. What is the chance of no storm?
(iii) \(P(\text{green}) = \dfrac{3}{10}\). Find \(P(\text{not green})\).
| \(1 - 0.15\) | \(=\) | \(0.85\) |
| \(100 - 40\) | \(=\) | \(60\) |
| \(1 - \dfrac{3}{10}\) | \(=\) | \(\dfrac{7}{10}\) |
(i) \(0.85\)
(ii) \(60\%\)
(iii) \(\dfrac{7}{10}\)
(i) \(P(\text{not chocolate})\)
(ii) \(P(\text{neither blueberry nor chocolate})\)
| \(1 - \dfrac{12}{36}\) | \(=\) | \(\dfrac{2}{3}\) |
| \(36 - 15 - 12\) | \(=\) | \(9\) |
| \(\dfrac{9}{36}\) | \(=\) | \(\dfrac{1}{4}\) |
(i) \(\dfrac{2}{3}\)
(ii) \(\dfrac{1}{4}\). Check: \(1 - \dfrac{27}{36} = \dfrac{1}{4}\).
| \(1 - \dfrac{3}{6}\) | \(=\) | \(\dfrac{1}{2}\) |
There are \(3\) A cards out of \(6\), so \(P(\text{not A}) = \dfrac{1}{2}\).
(i) How many red jelly beans are there?
(ii) Find \(P(\text{white})\).
(iii) Find \(P(\text{not white})\).
| \(\dfrac{2}{5} \times 60\) | \(=\) | \(24\) |
| \(60 - 24 - 15\) | \(=\) | \(21\) |
| \(\dfrac{21}{60}\) | \(=\) | \(\dfrac{7}{20}\) |
| \(1 - \dfrac{7}{20}\) | \(=\) | \(\dfrac{13}{20}\) |
(i) \(24\)
(ii) \(\dfrac{7}{20}\)
(iii) \(\dfrac{13}{20}\)
(i) Find \(P(\text{green})\).
(ii) There are \(40\) counters. How many are green?
| \(1 - \dfrac{2}{10} - \dfrac{3}{10}\) | \(=\) | \(\dfrac{5}{10}\) |
| \(\dfrac{1}{2} \times 40\) | \(=\) | \(20\) |
(i) \(\dfrac{1}{2}\)
(ii) \(20\)
Common pitfalls
Frequently asked questions
What is a complementary event?
All the outcomes that are not in the event. Together the event and its complement make up the whole sample space.
What do complementary probabilities add to?
\(1\) (or \(100\%\)).
How do you find the probability that something does not happen?
Take its probability away from \(1\): \(P(\text{not } A) = 1 - P(A)\).
If there is a 22% chance of rain, what is the chance of no rain?
\(100\% - 22\% = 78\%\).
Are “less than 3” and “more than 3” complementary on a die?
No. The \(3\) is in neither event, so together they do not cover every outcome.