The complement rule is a shortcut for finding the probability that an event does not happen. If an event A has some chance of occurring, then everything outside A is called not A or Aᶜ. Together, A and not A make up the entire sample space, so their probabilities add to 1.
This rule matters because many probability problems are easier to solve by counting what is missing instead of counting what is included.
The rule is written as P(not A) = 1 − P(A) or P(Aᶜ) = 1 − P(A). It is especially useful for at-least-one problems, where finding the probability of one or more successes can be harder than finding the probability of zero successes. For example, the probability of getting at least one head in several coin flips is 1 minus the probability of getting no heads.
The complement rule helps students turn complicated event descriptions into simpler calculations.
Key Facts
- P(not A) = 1 − P(A)
- P(Aᶜ) = 1 − P(A)
- P(A) + P(Aᶜ) = 1
- For at least one success, P(at least one) = 1 − P(none)
- If P(A) = 0.73, then P(Aᶜ) = 1 − 0.73 = 0.27
- An event and its complement cannot happen at the same time, but one of them must happen.
Vocabulary
- Event
- An event is a set of outcomes from a probability experiment.
- Complement
- The complement of an event is the set of all outcomes in the sample space that are not in the event.
- Sample space
- The sample space is the set of all possible outcomes of a probability experiment.
- Probability
- Probability is a number from 0 to 1 that measures how likely an event is to occur.
- At least one
- At least one means one or more occurrences of an event.
Common Mistakes to Avoid
- Subtracting from 100 instead of 1 when using decimal probabilities is wrong because probabilities in decimal form must add to 1, not 100.
- Treating A and Aᶜ as independent events is wrong because complements are directly linked and their probabilities always add to 1.
- Using P(at least one) = P(one) is wrong because at least one includes one, two, three, and all higher possible numbers of successes.
- Forgetting to define the complement clearly is wrong because solving the wrong opposite event leads to a wrong probability.
Practice Questions
- 1 A weather forecast says the probability of rain tomorrow is 0.35. What is the probability that it does not rain tomorrow?
- 2 A fair coin is flipped 4 times. Use the complement rule to find the probability of getting at least one head.
- 3 A student wants the probability that at least one person in a group has a birthday in July. Explain why it may be easier to use the complement rule than to count all cases with one or more July birthdays.