Algebra of Events

Two or more sets can be combined using four different operations, union, intersection, difference, and compliment. Since events are nothing but subsets of sample space, which means they are also set by themselves. In the same manner, two or more events can be combined using these operations. Let’s consider three events A, B, and C defined over the sample space S. 

Complimentary Event 

For every event A, there exists another event A’, which is called a complimentary event.  It consists of all those elements which do not belong to event A. For example, in the coin-tossing experiment. Let’s say event A is defined as getting one head. 

So, A = {HT, TH, HH} 

The complementary A’ of event A will be consists of all the elements in the sample space which are not in event A. Thus, 

A’ = {TT}

Event A or B

The Union of two sets A and B is denoted as A ∪ B. This contains all the elements which are in either set A, set B, or both. This event A or B is defined as, 

Event A or B = A ∪ B

OR

A ∪ B = {w : w ∈ A or w ∈ B}

Events A and B

The intersection of two sets A and B is denoted as A ∩ B. This contains all the elements which are in both set A and set B. This event A and B is defined as, 

Event A and B = A ∩ B

OR

A ∩ B= {w: w ∈ A and w ∈ B}

Event A but not B

The set difference A – B consists of all the elements which are in A but not in B. The events A but not B are defined as, 

A but not B = A – B 

OR

A – B = A ∩ B’

Where B’ is the complement of event B.

Using these concepts two other types of events are defined, that are:

• Mutually Exclusive Events
• Exhaustive Events

Let’s understand these two events as follows:

Mutually Exclusive Events

If the two events have nothing in common, then they are called mutually exclusive events, mutually exclusive events are similar to mutually exclusive sets. Formally two events A and B are called mutually exclusive if both of them cannot occur simultaneously. In this case, sets A and B are disjoint.

A ∩ B = ∅ 

For example, consider rolling a die, 

S = {1, 2, 3, 4, 5, 6} 

Now, event A is defined as “getting an even number” while event B is defined as “getting an odd number”. Now, these two events cannot occur together. 

A = {2, 4, 6} and B = {1, 3, 5}. 

Thus, the intersection between these two sets is an empty set. 

Exhaustive Events

Exhaustive events are those mutually exclusive events that together cover all the possible outcomes of an experiment. In other words, when all possible outcomes of an experiment are listed, and no other outcome exists outside of the list, it is said to be exhaustive. Formally we can define exhaustive events as let’s consider three events A, B, and C will be called exhaustive events if, 

A ∪ B ∪ C = S

In a more general setting, n events such that E₁, E₂,. . ., E_n is called exhaustive events if, 

 E₁ ∪ E₂ ∪. . .∪ E_n = S 

As an example, let’s say for a two-times coin toss experiment, 

A = Getting at least One head. 

B = Getting two tails. 

A = {HT, TH, HH} and B = {TT} 

Thus, A ∪ B = S

Read More,

• Chance and Probability
• Probability Theory
• Probability Distribution

Whenever an experiment is performed whose outcomes cannot be predicted with certainty, it is called a random experiment. In such cases, we can only measure which of the events is more likely or less likely to happen. This likelihood of events is measured in terms of probability and events refer to the possible outcomes of an experiment. Also, events can be classified into various different types based on different properties and probability values of events. 

In this article, we’ll explore the various types of events in probability, including simple events, compound events, mutually exclusive events, independent events, and dependent events. So, let’s dive into the world of different types of events.