Complement of an Event

Complement of an Event

The opposite of an event. That is, the set of all outcomes of an experiment that are not included in an event. The complement of event A is written A^C and is often read aloud as "not A".

$Formula: P(A^C) = 1 − P(A) or P(not A) = 1 − P(A). Example: Fair six-sided die, A={1,2}, A^C={3,4,5,6}, P(A)=1/3, P(A^C)=2/3.$

See also

Probability, complement of a set

Key Formula

P(A^C) = 1 - P(A)

Where:

$P(A)$ = The probability that event A occurs
$P(A^C)$ = The probability that event A does not occur (the complement)
$1$ = The total probability of all outcomes in the sample space

Worked Example

Problem: A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. Let event A be drawing a red marble. Find the probability of the complement of A (drawing a marble that is not red).

Step 1: Count the total number of marbles in the bag.

\text{Total} = 3 + 5 + 2 = 10

Step 2: Find the probability of event A (drawing a red marble).

P(A) = \frac{3}{10}

Step 3: Apply the complement formula to find the probability of not drawing a red marble.

P(A^C) = 1 - P(A) = 1 - \frac{3}{10} = \frac{7}{10}

Step 4: Verify: the complement includes blue and green marbles, which total 5 + 2 = 7 out of 10. This matches our answer.

P(A^C) = \frac{7}{10} = 0.7

Answer: The probability of not drawing a red marble is 7/10 or 0.7.

Another Example

This example uses a real-world probability given as a decimal rather than counting outcomes from a sample space. It shows the complement formula works even when you don't know the individual outcomes — you just need P(A).

Problem: The probability that it rains on any given day in a city is 0.15. What is the probability that it does not rain on a randomly chosen day?

Step 1: Identify event A as 'it rains,' with the given probability.

P(A) = 0.15

Step 2: The complement Aᶜ is 'it does not rain.' Apply the complement formula.

P(A^C) = 1 - 0.15 = 0.85

Step 3: Express the result as a percentage if needed.

P(A^C) = 85\%

Answer: The probability that it does not rain is 0.85, or 85%.

Frequently Asked Questions

What is the difference between the complement of an event and the complement of a set?

They are essentially the same idea applied in different contexts. The complement of a set refers to all elements in the universal set that are not in a given set. The complement of an event is that same concept applied specifically to probability: it is all outcomes in the sample space not included in the event. The formula P(Aᶜ) = 1 − P(A) is specific to probability.

Can the complement of an event be the empty set?

Yes. If event A is the entire sample space (meaning A is certain to happen, with P(A) = 1), then its complement is the empty set ∅, and P(Aᶜ) = 0. Similarly, if A is the empty set (an impossible event), then Aᶜ is the entire sample space.

When should you use the complement rule instead of calculating probability directly?

Use the complement rule when computing the probability of 'at least one' occurrence, or when the event itself has many favorable outcomes that are tedious to count. For example, finding P(at least one head in 10 coin flips) is much easier as 1 − P(no heads) than listing all the ways to get one or more heads.

Complement of an Event vs. Event Itself

	Complement of an Event	Event Itself
Definition	All outcomes where event A does NOT occur	The specific set of outcomes of interest
Notation	Aᶜ, A', or Ā	A
Probability relationship	P(Aᶜ) = 1 − P(A)	P(A) = 1 − P(Aᶜ)
Union of both	A ∪ Aᶜ = S (the entire sample space)	A ∪ Aᶜ = S (the entire sample space)
Intersection of both	A ∩ Aᶜ = ∅ (they are mutually exclusive)	A ∩ Aᶜ = ∅ (they are mutually exclusive)

Why It Matters

The complement rule is one of the most frequently used shortcuts in probability. You will encounter it in problems involving 'at least one' scenarios, such as finding the probability of at least one defective item in a batch or at least one success in repeated trials. It also appears throughout statistics, including hypothesis testing, where significance levels and their complements (confidence levels) are directly related by this rule.

Common Mistakes

Mistake: Confusing the complement with just one other outcome instead of ALL other outcomes.

Correction: When rolling a die, the complement of 'rolling a 6' is not just 'rolling a 1' — it is 'rolling a 1, 2, 3, 4, or 5.' The complement includes every outcome in the sample space that is not in the original event.

Mistake: Forgetting that P(A) + P(Aᶜ) must equal exactly 1.

Correction: If you calculate P(A) = 0.4 and P(Aᶜ) = 0.5, something is wrong. Always check that the two probabilities sum to 1. This is a reliable way to catch arithmetic errors.

Related Terms

Event — The original set whose complement is taken
Outcome — Individual results that make up events and complements
Experiment — The process that generates the sample space
Probability — Numerical measure applied to events and complements
Complement of a Set — The general set theory concept behind this rule
Sample Space — The universal set from which complements are defined
Mutually Exclusive Events — An event and its complement are always mutually exclusive