Events in Probability
Events in probability can be defined as a set of outcomes of a random experiment. The sample space indicates all possible outcomes of an experiment. Thus, events in probability can also be described as subsets of the sample space.
There are many different types of events in probability. Each type of event has its own individual properties. This classification of events in probability helps to simplify mathematical calculations. In this article, we will learn more about events in probability, their types and see certain associated examples.
What are Events in Probability?
Events in probability are outcomes of random experiments. Any subset of the sample space will form events in probability. The likelihood of occurrence of events in probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes of that experiment.
Definition of Events in Probability
Events in probability can be defined as certain likely outcomes of an experiment that form a subset of a finite sample space. The probability of occurrence of any event will always lie between 0 and 1. There could be many events associated with one sample space.
Events in Probability Example
Suppose a fair die is rolled. The total number of possible outcomes will form the sample space and are given by {1, 2, 3, 4, 5, 6}. Let an event, E, be defined as getting an even number on the die. Then E = {2, 4, 6}. Thus, it can be seen that E is a subset of the sample space and is an outcome of the rolling of a die.
Types of Events in Probability
There are several different types of events in probability. There can only be one sample space for a random experiment however, there can be many different types of events. Some of the important events in probability are listed below.
Independent and Dependent Events
Independent events in probability are those events whose outcome does not depend on some previous outcome. No matter how many times an experiment has been conducted the probability of occurrence of independent events will be the same. For example, tossing a coin is an independent event in probability.
Dependent events in probability are events whose outcome depends on a previous outcome. This implies that the probability of occurrence of a dependent event will be affected by some previous outcome. For example, drawing two balls one after another from a bag without replacement.
Impossible and Sure Events
An event that can never happen is known as an impossible event. As impossible events in probability will never take place thus, the chance that they will occur is always 0. For example, the sun revolving around the earth is an impossible event.
A sure event is one that will always happen. The probability of occurrence of a sure event will always be 1. For example, the earth revolving around the sun is a sure event.
Simple and Compound Events
If an event consists of a single point or a single result from the sample space, it is termed a simple event. The event of getting less than 2 on rolling a fair die, denoted as E = {1}, is an example of a simple event.
If an event consists of more than a single result from the sample space, it is called a compound event. An example of a compound event in probability is rolling a fair die and getting an odd number. E = {1, 3, 5}.
Complementary Events
When there are two events such that one event can occur if and only if the other does not take place then such events are known as complementary events in probability. The sum of the probability of complementary events will always be equal to 1. For example, on tossing a coin let E be defined as getting a head. Then the complement of E is E' which will be the event of getting a tail. Thus, E and E' together make up complementary events. Such events are mutually exclusive and exhaustive.
Mutually Exclusive Events
Events that cannot occur at the same time are known as mutually exclusive events. Thus, mutually exclusive events in probability do not have any common outcomes. For example, S = {10, 9, 8, 7, 6, 5, 4}, A = {4, 6, 7} and B = {10, 9, 8}. As there is nothing common between sets A and B thus, they are mutually exclusive events.
Exhaustive Events
Exhaustive events in probability are those events when taken together from the sample space of a random experiment. In other words, a set of events out of which at least one is sure to occur when the experiment is performed are exhaustive events. For example, the outcome of an exam is either passing or failing.
Equally Likely Events
Equally likely events in probability are those events in which the outcomes are equally possible. For example, on tossing a coin, getting a head or getting a tail, are equally likely events.
Intersection of Events in Probability
The intersection of events in probability corresponds to the AND event. If two events are associated with the "AND" operator, it implies that the common outcomes of both events will be the result. It is denoted by the intersection symbol "∩". For example, A = {1, 2, 3, 4}, B = {2, 3, 5, 6} then A ∩ B = {2, 3}.
Union of Events in Probability
The union of events in probability is the same as the OR event. If there are two events that belong to this group then the outcomes of either event or both will be the result. The union symbol (∪) is used to denote the OR event. For example, A = {1, 2, 3, 4}, B = {2, 3, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6}.
How to Find Probability of an Event?
To find the likelihood of occurrence of events in probability the steps are as follows:
 Determine the sample space or the total number of possible outcomes of the experiment.
 Determine the number of favorable outcomes of the event.
 Divide the value from step 2 by the value obtained in step 1 to get the required probability.
Related Articles:
 Probability and Statistics
 Probability Rules
 Conditional Probability Formula
 Experimental Probability
 Event Probability Calculator
Important Notes on Events in Probability
 Events in probability can be defined as certain outcomes of a random experiment.
 Events in probability are a subset of the sample space.
 The types of events in probability are simple, sure, impossible, complementary, mutually exclusive, exhaustive, equally likely, compound, independent, and dependent events.
Examples on Events in Probability

Example 1: A random card is drawn from a deck of 52 cards. What is the probability that it is an ace?
Solution: E = event of drawing an ace.
Total number of outcomes = 52
The favorable number of outcomes = 4 (there are 4 ace cards in a deck of cards. One belonging to each suit).
P(E) = 4 / 52 = 1 / 13
Answer: P(E) = 1 / 13 
Example 2: On rolling a fair dice, A is the event of getting a number less than 5, B is the event of getting an odd number and C is the event of getting a multiple of 3. Find the AND event.
Solution: Sample space of a dice roll = {1, 2, 3, 4, 5, 6}
A = {1, 2, 3, 4}
B = {1, 3, 5}
C = {3, 6}
A ∩ B ∩ C = {3}
Answer: A ∩ B ∩ C = {3} 
Example 3: If a coin is tossed 3 times what would be the event of getting at most two heads?
Solution: The sample space for tossing coin thrice is {(H, H, H), (T, H, H), (H, T, H), (H, H, T), (T, T, H), (H, T, T), (T, H, T), (T, T, T)}
E = {(T, H, H), (H, T, H), (H, H, T), (T, T, H), (H, T, T), (T, H, T), (T, T, T)}
Answer: The event of getting at most two heads is {(T, H, H), (H, T, H), (H, H, T), (T, T, H), (H, T, T), (T, H, T), (T, T, T)}
FAQs on Events in Probability
What is Meant by Events in Probability?
Events in probability refer to certain outcomes of a random experiment that form a part of the sample space. The probability of occurrence of any event will lie between 0 and 1.
What are the Types of Events in Probability?
The various types of events in probability are listed below:
 Independent events
 Dependent events
 Simple events
 Compound events
 Impossible events
 Sure events
 Complementary events
 Mutually exclusive events
 Exhaustive events
 Equally likely events
What are Independent and Depending Events in Probability?
Independent events are those events that do not depend on some previous outcome while dependent events get affected by previous outcomes.
What is the Likelihood of Occurrence of Impossible Events in Probability?
Impossible events in probability are events that can never happen. Thus, the probability of occurrence of such events will always be 0.
What is the Difference Between Mutually Exclusive and Complementary Events in Probability?
Mutually exclusive events are events that cannot occur at the same time, however, these events combined need not fill up the sample space. Complementary events in probability are mutually exclusive events that are also exhaustive.
What are Simple Events in Probability?
Simple events in probability are events that have a single point only. Such events include only one result from the sample space.
What are Compound Events in Probability?
Events that can have more than one result from the sample space are known as compound events in probability.
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