What does it mean when two events are complements?

Verified step by step guidance

1. Understand that in probability, two events are complements if they represent all possible outcomes between them, with no overlap.
2. Formally, if event A is one event, its complement, denoted as A^c, consists of all outcomes not in A.
3. The key property of complementary events is that their probabilities add up to 1, which can be written as: \(P(A) + P(A^c) = 1\).
4. This means that if you know the probability of event A occurring, you can find the probability of its complement by subtracting from 1: \(P(A^c) = 1 - P(A)\).
5. Complementary events are mutually exclusive and collectively exhaustive, meaning they cannot happen at the same time, and together they cover all possible outcomes.

Key Concepts

Complementary Events

Complementary events are two outcomes of a random experiment that are mutually exclusive and exhaustive, meaning one event occurs if and only if the other does not. Together, they cover all possible outcomes, so their probabilities add up to 1.

Mutually Exclusive Events

Mutually exclusive events cannot happen at the same time. If one event occurs, the other cannot. This property is essential for understanding complements because complementary events must be mutually exclusive.

Probability of an Event and Its Complement

The probability of an event and its complement always sums to 1. If the probability of an event A is P(A), then the probability of its complement A' is 1 - P(A), representing the chance that event A does not occur.