Complement Rule
Compute the probability that something never happens, then subtract from 1.
What is it?
. When is "at least one", its complement is "none", which is often a single product.
Why is it useful?
"At least one" questions have many cases directly but only one case via the complement.
When should you think of it?
Use it when the event has many ways to occur but its opposite has very few.
Think of this method when…
- Wording "at least one", "at least two share", "ever"
- Direct counting needs many overlapping cases
- The opposite event is "none" or "all different"
- Independent trials where the complement is a product
Core idea
Count the easy opposite event and subtract.
Step-by-step playbook
- 1State the complement in words.
- 2Compute its probability, often as a product.
- 3Subtract from 1.
Worked example
Worked example
A fair die is rolled 4 times. What is the probability of at least one six?
- 1
Complement: no six in 4 rolls, probability .
- 2
Subtract.
- Answer:
Common mistakes
- Adding probabilities of each trial (), which double-counts.
- Negating the event incorrectly (the complement of "at least two" is "zero or one").
Would you use this method?
With 23 people, what is the chance two share a birthday?
Practice this method
17 QuantSolve questions are tagged with this method.
Practice this method