Complementary Events: Definition, Rule, and Examples

Complementary events are fundamental concepts in probability theory that provide insights into the relationship between different outcomes of an experiment or event. In probability, events are not isolated occurrences but often have complementary counterparts that represent the opposite or negation of the original event.

What is Probability?

Probability theory is a branch of mathematics that deals with the interpretation of random events and the likelihood of these events occurring. Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1.

The values of probability range from 0 to 1, with 0 telling us improbability and 1 denoting certainty.

For calculating probability, we simply divide the number of favourable outcomes by the total number of outcomes.

What are Complementary Events in Probability?

Complementary events play a crucial role in probability theory, providing a fundamental concept for analyzing the likelihood of outcomes in various scenarios. Understanding complementary events is essential for solving probability problems and making informed decisions in fields ranging from statistics to finance and beyond.

Complementary Events Definition

Complementary events are two mutually exclusive events where the occurrence of one event implies the non-occurrence of the other event, and vice versa. In other words, if event A represents a certain outcome, then the complementary event, denoted as A’, represents the outcome opposite to A.

Complementary Events

For example, if event A is rolling a 3 on a fair six-sided die, then the complementary event A’ is rolling any number other than 3.

Complementary Events Properties

• Mutual Exclusivity: Complementary events cannot occur simultaneously. If one event happens, the other event cannot happen at the same time.

• Complete Coverage: Together, complementary events cover all possible outcomes of an experiment. Either event A or its complementary event A’ must occur.

• Probability Sum: Sum of the probabilities of complementary events is always equal to 1. If the probability of event A occurring is P(A), then the probability of its complementary event A’ occurring is P(A′) = 1 − P(A).

Rule of Complementary Events

Rule of complementary events provides a straightforward way to calculate the probability of one event by subtracting the probability of its complementary event from 1. Mathematically, it can be expressed as:

P(A′) = 1 – P(A)

where:

• P(A′) is Probability of Complementary event A’.
• P(A) is Probability of Event A.

Complementary Events Example

Suppose we toss a fair coin. Let event A represent getting heads on the coin toss. The complementary event A’ would then represent getting tails. Since the coin is fair, the probability of getting heads (P(A)) is 0.5. Using the rule of complementary events, we can find the probability of getting tails (P(A′)) as follows:

P(A′) = 1 – P(A)

= 1 − 0.5 = 0.5

Therefore, the probability of getting tails is also 0.5, as expected.

Sample Questions on Complementary Events

Question 1: A standard deck of playing cards contains 52 cards, with 13 cards in each suit (hearts, diamonds, clubs, spades). If you draw a card at random from the deck, what is the probability that it is not a heart?

Solution:

Since there are 13 hearts in a standard deck of 52 cards,

Probability of drawing a heart is,

13/52 = 1/4

Therefore, the probability of not drawing a heart (drawing any other suit) is,

1 – 1/4 = 3/4

Question 2: A fair six-sided die is rolled. What is the probability of not rolling ‘1’?

Solution:

Since there is 1 outcome corresponding to rolling ‘1’ and ‘6’ possible outcomes in total when rolling a fair six-sided die,

Probability of rolling ‘1’ is 1/6

Therefore, the probability of not rolling ‘1’ is,

1 – 1/6 = 5/6

Question 3: You toss a fair coin three times. What is the probability of getting at least one head?

Solution:

Complementary event of getting at least one head is getting no heads at all (getting all tails)

Since each toss of a fair coin is independent, the probability of getting tails on a single toss is 1/2

Therefore, the probability of getting all tails in three tosses is

(1/2)³ = 1/8

Hence, the probability of getting at least one head is

1 – 1/8 = 7/8

Question 4: A bag contains 10 red marbles and 5 blue marbles. If you randomly select a marble from the bag, what is the probability that it is not red?

Solution:

Since there are 10 red marbles and 15 marbles in total in the bag, the probability of selecting a red marble is

10/15 = 2/3

Therefore, the probability of not selecting a red marble (selecting a blue marble) is

1 – 2/3 = 1/3

Question 5: In a class of 30 students, 20 students are studying mathematics. If a student is selected at random from the class, what is the probability that the student is not studying mathematics?

Solution:

Since there are 20 students studying mathematics and 30 students in total in the class, the probability of selecting a student studying mathematics is

20/30 = 2/3

Therefore, the probability of not selecting a student studying mathematics is

1 – 2/3 = 1/3

Summary

In conclusion, complementary events are essential in probability theory for analyzing the likelihood of outcomes and making informed decisions. By understanding the definition, properties, rule, and examples of complementary events, one can effectively apply them to solve various probability problems in real-world scenarios.

Frequently Asked Questions on Complementary Events

What are complementary events in probability?

Complementary events are two events that are mutually exclusive and cover all possible outcomes of an experiment. If one event occurs, the other cannot occur, and vice versa. For example, when rolling a fair six-sided die, the event of rolling a 1 and the event of not rolling a 1 are complementary.

How do you find the probability of a complementary event?

To find the probability of a complementary event, subtract the probability of the original event from 1. If the probability of event A is P(A), then the probability of its complementary event A’ is 1 – P(A).

What is the significance of complementary events in probability calculations?

Complementary events play a crucial role in probability calculations by providing an alternative perspective. They allow us to simplify complex probability problems and make calculations more manageable. Additionally, complementary events help ensure that all possible outcomes are accounted for in probability analysis.

Can complementary events occur simultaneously?

No, complementary events cannot occur simultaneously. If one event happens, the other event cannot happen at the same time. For example, when flipping a coin, the events of getting heads and getting tails are complementary, and only one of them can occur on each flip.

How do complementary events relate to the rule of complements?

The rule of complements states that the sum of the probabilities of an event and its complementary event is always equal to 1. This rule highlights the relationship between complementary events and underscores their importance in probability theory.