The Lesson

The complement of an event in probability is all outcomes that are not the event.

A Real Example of Complementary Events

It is easier to understand complementary events with an example.

Tossing a Coin

If a coin is tossed, the coin can land on Heads.

The complement to this event is the event where the coin does not land on Heads. If the coin does not land on Heads, it must land on Tails.

Tails is the event not Heads. Tails is the complement to the event Heads. Heads is the event not Tails. Heads is the complement to the event Tails. Together, Heads and Tails are complementary events for the toss of a coin.

The Complement Rule

The probability of complementary events sum to 1:
Probability of event happening + Probability of event not happening = 1
Therefore:
Probability of event happening = 1 − Probability of event not happening
Probability of event not happening = 1 − Probability of event happening

A Note on Notation

If an event is A, then we can write the complement as not A, A', Ac or Ā. The probability of A can be written as P(A) The probability of the complement of A can be written as P(not A), P(A'), P(Ac) or P(Ā).

Complementary Events Are Mutually Exclusive and Exhaustive

Complementary events are mutually exclusive and exhaustive events. Consider a coin toss.
  • Heads and Tails cannot happen at the same time. They are mutually exclusive events.
  • Heads and Tails are all of the possible outcomes for a coin toss. They are exhaustive events.