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Whenever an experiment is performed whose outcomes cannot be predicted with certainty, it is called a random experiment. For example, while tossing a coin no one can predict the outcome — will it be heads or tails? 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. The outcomes are called events, for example in the experiment mentioned above getting a head or tail is called an event. Events can also be classified into different categories. Let’s study those categories in detail.

### Events and Sample Spaces

When an experiment is performed, there are outcomes that are possible but none of them can be predicted. In the field of probability and statistics. It becomes essential to list down the total possible outcomes of any experiment. Consider an experiment of tossing a coin two times, the figure demonstrates the number of possibilities in such trials. From the figure, it is obvious there are four possible outcomes. Sample Space is the set of total possible outcomes.

S = {HH, HT, TH, TT}

Events

An event is described as a set of outcomes. For example, getting a tail in a coin toss is an event while all the even-numbered outcomes while rolling a die also constitutes an event.

An event is a subset of the sample space.

Occurrence of an Event

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.

So, an event E of a sample space S is said to have occurred if the outcome w of the experiment is such that w ∈ E. When an outcome is such that it does not belong to the set E. It is said to have not occurred.

### Type of Events

It is now clear that events are subsets of 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 elements they have. The following list gives the different types of events:

1. Impossible and Sure Events
2. Simple Event
3. Compound Event

Let’s see them one by one.

Impossible and Sure Events

To get an intuition for this type of event, 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.

Since, these kinds of events are impossible and can be described by an empty set ∅. These are called impossible and sure events.

Simple Event

When an event consists of only one point of the sample space, this event is called a simple event.

Considering the earlier example of two coin tosses, the sample space for that experiment is,

S = {HH, HT, TH, TT}

Event “E” is defined as getting two tails. So, in the sample space, there is only one outcome where it happens. Thus, this is an example of a simple event.

Compound Event

Events that have more than one sample point are called compound events. Consider the previous example and define event E2 as the getting one head. Thus, the event will consist of three points of sample space.

E2 = {HH, HT, TH}

Thus, this is called a compound event.

### 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

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

= {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

= {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

= A ∩ B’

Using these concepts two other types of events are defined. Let’s look at them.

Mutually Exclusive Events

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 of 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, intersection between these two sets is an empty set.

Exhaustive Events

Events A, B, and C will be called exhaustive events if,

A ∪ B ∪ C = S

In a more general setting, events E1, E2 …….En is called exhaustive events if,

E1 ∪ E2 …. ∪ En = 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

### Axiomatic Approach to Probability

For a random experiment, let S define a sample space. The probability P is a real-valued function whose domain is the power set of S and range is between [0, 1]. Intuitively, it measures the chances of happening some event. The probability of any event must satisfy these axioms:

1. For any event E, P(E) ≤ 1.
2. P(S) = 1
3. If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F)

The third axiom can be generalized to n number of events,

P(E1 ∪ E2 …. ∪ En) = P(E1) + P(E2) + .. P(En

### Sample Problems

Question 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?

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.

Question 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.”

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.

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

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

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

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}

= {6}

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

= ∅ (Empty Set)

A ∪ B = {4, 5, 6} ∪ {3, 6}

= {3, 4, 5, 6}

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

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}

= {2}

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

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