What is a complement?
The complement of an event A is the event that A does not happen. We write it as A′ (read as "A prime" or "A complement"). For example, if A is "rolling a 6" on a die, then A′ is "not rolling a 6", which means rolling a 1, 2, 3, 4 or 5.
Key idea
Because an event either happens or it does not, P(A) + P(A′) = 1. This gives the complement rule: P(A′) = 1 − P(A).
Why the rule helps
Sometimes it is much easier to work out the probability that something does not happen, and then subtract from 1. The complement rule turns a hard counting problem into a quick subtraction.
Worked example
A fair die is rolled. Find the probability of not getting a 6.
P(getting a 6) = 1 ÷ 6.
P(not getting a 6) = 1 − 1/6 = 6/6 − 1/6 = 5 ÷ 6.
Working with decimals
The complement rule also works with decimals and percentages. If the probability that it rains tomorrow is 0.7, then the probability that it does not rain is 1 − 0.7 = 0.3. The two probabilities must always add up to 1.
Remember
- P(A) + P(A′) = 1 always.
- A′ means "A does not happen".
- Subtract from 1, whether the probability is a fraction, decimal or percentage.