Complementary Events

In probability theory, an event is a set of outcomes of an experiment to which a probability is assigned. When dealing with a probability space, the set of all possible outcomes is called the sample space, often denoted by the symbol $S$. A complementary event is a concept that is fundamental to the understanding of probability.

Definition

The complementary event of an event $A$ is the event that $A$ does not occur. It includes all outcomes of the sample space that are not in $A$. If $A$ is an event, the complement of $A$ is denoted by $A'$, $A^c$, or $\overline{A}$.

Properties

The main properties of complementary events are:

1. The probability of an event and its complement always sum to 1.
2. The complement of the complement of an event is the event itself.

Mathematically, these properties can be expressed as:

1. $P(A) + P(A') = 1$
2. $(A')' = A$

Formulas

The probability of the complement of an event $A$ can be calculated using the formula:

$$ P(A') = 1 - P(A) $$

This formula is derived from the first property of complementary events.

Examples

Let's consider a simple example to illustrate the concept of complementary events:

Example 1: Coin Toss

When you toss a fair coin, there are two possible outcomes: heads (H) or tails (T). The sample space is $S = {H, T}$. Let's define event $A$ as getting heads. Then, the complementary event $A'$ is getting tails.

• $P(A) = \frac{1}{2}$ (since there is one head out of two possible outcomes)
• $P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$

Example 2: Rolling a Die

Consider rolling a fair six-sided die. The sample space is $S = {1, 2, 3, 4, 5, 6}$. Let's define event $B$ as rolling an even number. Then, the complementary event $B'$ is rolling an odd number.

• $P(B) = \frac{3}{6} = \frac{1}{2}$ (since there are three even numbers: 2, 4, 6)
• $P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2}$

Table of Differences and Important Points

Aspect	Event $A$	Complementary Event $A'$
Definition	Set of outcomes where $A$ occurs	Set of outcomes where $A$ does not occur
Probability	$P(A)$	$P(A') = 1 - P(A)$
Relationship	$P(A) + P(A') = 1$	$P(A) + P(A') = 1$
Complement of Complement	$(A')' = A$	$(A')' = A$

Conclusion

Understanding complementary events is crucial for solving probability problems. It simplifies calculations and helps in understanding the structure of a probability space. Remember that the sum of the probabilities of an event and its complement is always equal to 1, which is a fundamental aspect of probability theory.