## 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'**,

**A**or

^{c}**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(A**or

^{c})**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.