Types of Events in Probability

Events refer to the possible outcomes of a random experiment. Where experiment is a process that generates well-defined outcomes under specific conditions. Likelihood of events is measured in terms of probability, that is, which of the events is more likely or less likely to happen. Also, events can be classified into various types based on different properties and probability values of events. 

An event is a subset of the sample space.

For example:

• Experiment: Coin toss, Sample sapce = {'head' , 'tail'}, Event = "getting a tail"
• Experiment: Rolling a die, Sample space: {1,2,3,4,5,6}, Event="the even-numbered outcomes"

Consider an experiment of throwing a die. Let's say that event E is defined as "getting an even number". So, if a number 4 comes up, it is said that event E has occurred.

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 it.

Type of Events

It is now clear that events are subsets of the sample space. It is essential to understand the difference between different types of events that can happen while performing random experiments. This understanding of events helps us in calculating the probabilities for both simple and complex random experiments. We know that events are basically set, so they can be classified on the basis of the elements they have. The following list gives the different types of events: 

Let's see them one by one.

Impossible Event 

These kinds of events are impossible and can be described by an empty set ∅ and are called Impossible Events.

Example:

Consider an experiment in which we roll a die. Now let's define an event that consists of outcomes that are multiple of 7. Sample space for this event is denoted by S, 

S = {1, 2, 3, 4, 5, 6} 

Now since there is no outcome in the sample space which is a multiple of 7. So, the set of event E will be an empty set. 

Sure Event

Such an event, which has a probability of 1, i.e., occurrence of the event is certain or universal truth then that event is called Sure Event or Certain Event.

Note: An event that consists of the entire sample space is called a sure event (or certain event).

Example:

If we roll a die, as the event is the occurrence of a number less than 7, then it is sure that the occurring number is always less than 7 as the die only has numbers 1, 2, 3, 4, 5, and 6. 

Simple Event

Any event that comprises a single result from the sample space is known as a simple event. 

Example:

The Sample space of rolling a die, S= {1, 2, 3, 4, 5, 6} and the event for getting less than 2, E= {1}, where E has a single result taken from the sample space, Hence the event is a Simple event.

Compound Event

A Compound event is just opposite to what a simple event is, that is, any event that comprises more than a single result or more than a single point from the sample space, that event is known as a Compound event. 

Example:

S={1, 2, 3, 4, 5, 6} and E= {3, 4, 5}, where E is a Compound event.

Dependent Events

Dependent events are those in which the next outcome depends on the previous outcomes, which means, the probability of an event will change based on its previous outcomes.

Example:

Experiment: "Drawing balls from a bag consisting of 4 black and 3 red balls"

Event: "Drawing a black ball from the bag without replacement"

1. First draw.
The probability of drawing a black ball = 4/7= 0.571.

2. After the first draw, (if a black ball was drawn)
Now the bag consist of 3 black and 3 red balls.
Now the probability of getting black ball is 3/6= 0.5

Thus, this event is dependent as the probability of each successive event depends upon the previous event.

Note : Here, there is a way of converting this dependent event into independent event, it can be done through Replacement. If after each experiment the ball is again kept in the bag, the sample space of the experiment will not change and hence, the probability of the event will remain same too.

Independent Event

Independent events are those in which the next outcome is independent of the previous outcome. This means the probability of the occurrence of an event will remain the same no matter how many times the same experiment is done. 

Example:

Experiment: Rolling a die,
Event: "getting an even number"
1. Rolling the die : P(getting an even number) = 0.5,
2. Now the dice is rolled again, still P(getting an even number) = 0.5.
This means, that the probability of the event is independent of its previous outcomes.

Equally Likely Events

Those outcomes of an experiment that have the same probability are called Equally Likely Events. In other words, if two or more events have the same likelihood of happening, they are considered equally likely events.

Example:

Consider rolling a fair six-sided die. Each of the six possible outcomes (1, 2, 3, 4, 5, and 6) has the same probability of occurring, which is 1/6.
Therefore, rolling a 1 is equally likely as rolling a 2, 3, 4, 5, or 6.

Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. If one happens, the other cannot.

Example:

When you roll a die:

• Event A: Getting a 2.
• Event B: Getting a 5.

These two events are mutually exclusive because you cannot get both 2 and 5 on the same roll.

Exhaustive Events

Exhaustive events are a set of events that cover all possible outcomes of an experiment. This means at least one of them must happen.

Example:

Tossing a coin:

• Event A: Getting Head.
• Event B: Getting Tail.

These two events are exhaustive because together they include all possible outcomes of the coin toss.

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 the given types of events can be explained in depth:

• Mutually Exclusive Events
• Exhaustive Events

Let's understand these two events as follows:

Mutually Exclusive Events: 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: 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 

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

Problems on Types of Events

Problems 1: Consider the experiment of tossing a fair coin 3 times, Event A is defined as getting all tails. What kind of event is this? 

Solution: 

Sample space for the coin toss will be, 

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

For the event A, 

A = {TTT}

This event is only mapped to one element of sample space. Thus, it is a simple event. 

Problems 2: Let's say a coin is tossed once, state whether the following statement is True or False. 

“If we define an event X which means getting both heads and tails. This event will be a simple event.”

Solution:

When a coin it tossed, there can be only two outcomes, Heads or Tails. 

S = {H, T} 

Getting both Heads and Tails is not possible, thus event X is an empty set. 

Thus, it is an impossible and sure event. So, this statement is False. 

Problems 3: A die is rolled, and three events A, B, and C are defined below:

1. A: Getting a number greater than 3 
2. B: Getting a number that is multiple of 3. 
3. C: Getting an odd number

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

Solution:

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {4, 5, 6}

For the event B, 

B = {3, 6}

For the event C, 

C = {1, 3, 5}

A ∩ B = {4, 5, 6} ∩ {3, 6}

⇒ A ∩ B = {6}

A ∩ B ∩ C = {4, 5, 6} ∩ {3, 6} ∩ {1, 3, 5}

⇒ A ∩ B ∩ C = ∅ (Empty Set) 

A ∪ B = {4, 5, 6} ∪ {3, 6}

⇒ A ∪ B = {3, 4, 5, 6}

Problems 4: A die is rolled, let's define two events, event A is getting the number 2 and Event B is getting an even number. Are these events mutually exclusive? 

Solution: 

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {2}

For the event B, 

B = {2, 4, 6}

For two events to be mutually exclusive, their intersection must be an empty set 

A ∩ B = {2} ∩ {2, 4, 6}

⇒ A ∩ B  = {2}

Since it is not an empty set, these events are not mutually exclusive.

Related Articles:

• Chance and Probability
• Probability Theory
• Probability Distribution