Types of Events: Probability is a numerical value that expresses the degree of our conviction in the occurrence of an event and is defined as a quantitative measure of uncertainty. An event’s probability is always a value inclusive, between \(0\) and \(1\). The closer an event’s probability is to \(1\), the more likely it will occur; the closer an event’s probability is to \(0\), the less likely it will occur. The likelihood of an event occurring is \(0\) if it cannot happen. Its possibility is \(1\) if it must occur, i.e., its occurrence is certain.

People frequently say things like “It might rain today” or “I will most likely pass the exam because it was not too difficult” or “Most likely, he will be selected” in everyday conversations. Words like possibly, probably, and most likely are used in all three statements to suggest the likelihood or certainty of something happening.

In this article, we will discuss the event and types of events in detail. Continue reading to know more.

Random Experiment

An experiment is random when the experiment has more than one possible outcome, and It is also impossible to predict with certainty which outcome will occur. In a coin-tossing experiment, for example, it is possible to predict with certainty whether the coin will land heads up or tails up, but not whether the coin will land heads up or tails up. Any of the six numbers \((1,2,3,4,5,6)\) may appear if a die is thrown once; nevertheless, it is impossible to anticipate which number will appear.

Outcome

The outcome is the possible outcome of a random experiment.

If you toss a coin twice, for example, some of the possible results are \(H H, H T\), and so on.

Sample Space

The set of all possible outcomes of an experiment is termed sample space. It is the universal set \(S\) unique to a given experiment. For the experiment of tossing a coin twice, the sample space is given by

\(S = \left\{{HH,HT,TH,TT} \right\}\)

The set of all cards in the deck serves as the sample space for the drawing cards experiment.

Event

A set of outcomes of an experiment is defined as a probability event. In other words, in probability, an event is the subset of the sample space.

Probability is defined as the likelihood of the occurrence of an event. Any event has a chance of occurring between \(0\) and \(1\).

The sample space for simultaneously tossing three coins is provided by:

$$\begin{gathered} S=\left\{(T,T,T),(T,T,H),(T,H,T),(T,H,H), \\ (H,T,T),(H,T,H),(H,H,T),(H,H,H)\right\} \end{gathered}$$

For example, if we only wish to identify outcomes with at least two heads, the set of all such possibilities can be written as:
\(E = \left\{{\left({H,T,H} \right),\left({H,H,T} \right),\left({H,H,H} \right)\left({T,H,H} \right)}  \right\}\)

As a result, an event is a subset of the sample space, i.e., \(E\) is a subset of \(S\). A particular sample space may contain many events. For any event to occur, the experiment’s result must be part of the collection of events \(E\).

What is the Probability of Occurrence of an Event?

The probability of any event is defined as the ratio of number of favourable outcomes to total number of outcomes.

As a result, the likelihood of an event occurring is given as:

\(P(E)=\frac{\text { Number of Favourable Outcomes }}{\text { Total Number of Outcomes }}\)

Types of Events

The following are some of the important probability events:

1. Impossible and Sure Events

An event is called an impossible event if the probability of occurrence of an event is \(0\). An event is called a sure event if the probability of occurrence of an event is \(1\). Alternatively, the empty set \(\varphi\) is an impossible event, and the sample space \(S\) is a sure event.

2. Simple or Elementary Events

Any event consisting of a single point of the sample space is a simple event in probability.

For example, if \(S = \left\{{5,78,96,54,89} \right\}\) and \(E = \left\{{78} \right\}\) then \(E\) is a simple event.

3. Compound Events

In contrast to the simple event, every event that involves more than one single point of the sample space is referred to as a compound event.

Using the same example as before, if \(S = \left\{{5,78,96,54,89} \right\}, {E_1} = \left\{{56,54} \right\},{E_2} = \left\{{78,56,89} \right\}\) then, \(E_{1}\) and \(E_{2}\) represent two compound events.

4. Independent Events and Dependent Events

If any event is completely unaffected by any other event, such events are known as independent events in probability.

Example: You toss a coin three times, and it comes up “heads” each time. What is the chance that the next toss will also be a “Head”?
The chance is simply \(\frac{1}{2}\) or \(50 \%\), just like any other toss of the coin.
What it did in the past will not affect the current toss!

The events which are affected by other events are known as dependent events. After taking one card from the deck, there are fewer cards available, so the probabilities change!

5. Mutually Exclusive Events

Two events \(A\) and \(B\) of a sample space \(S\) are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events \(A\) and \(B\) cannot occur simultaneously, and thus \(P(A \cap B)=0\).

For example, the elementary events \(\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ 4 \right\},\left\{ 5 \right\}\) or \(\left\{6 \right\}\) of the experiment of throwing dice are mutually exclusive. Consider the experiment of throwing a die once. The events \(E=\) getting an even number and \(F=\) getting an odd number are mutually exclusive events because of \(E \cap F=\varphi\).

6. Exhaustive Events

If \(E_{1}, E_{2}, \ldots, E_{n}\) are \(n\) events of a sample space \(S\) and if
\(E_{1} \cup E_{2} \cup E_{3} \ldots \cup E_{n}=U_{i=1}^{n} E_{i}=S\)
then \(E_{1}, E_{2}, \ldots, E_{n}\) are called exhaustive events.

In other words, events \(E_{1}, E_{2}, \ldots, E_{n}\) of a sample space \(S\) are said to be exhaustive if at least one of them necessarily occurs whenever the experiment is performed.

Take, for example, the act of rolling a die. We have \(S = \left\{{1,2,3,4,5,6} \right\}\).

Define the two events \(A\) : ‘a number less than or equal to \(4\) appears.’
\(B\) : ‘a number greater than or equal to \(4\) appears.’
Now \(A = \left\{{1,2,3,4} \right\},B = \left\{{4,5,6} \right\}\)
\(A \cup B = \left\{{1,2,3,4,5,6} \right\} = S\)
Such events \(A\) and \(B\) are called exhaustive events.

7. Complementary Events

Given an event \(A\), the complement of \(A\) is the event consisting of all sample space outcomes that do not correspond to the occurrence of \(A\).
The complement of \(A\) is denoted by \(A^{\prime}\) or \(A\). It is also called the event ‘not \(A^{\prime}\). Further \(P(A)\) denotes the probability that \(A\) will not occur.
\(A’ = A = S – A = \{ w:w \in S\) and \(w \notin A\} \)

8. Events Associated with “OR”

If two events \(E_{1}\) and \(E_{2}\) are associated with OR, then it means that either \(E_{1}\) or \(E_{2}\) or both. The union symbol \(\left(\cup  \right)\) is used to represent OR in probability.
Thus, the event \(E_{1} \cup E_{2}\) denotes \(E_{1}\) OR \(E_{2}\).

If we have mutually exhaustive events \(E_{1}, E_{2}, E_{3}, \ldots \ldots \ldots \ldots, E_{n}\) associated with sample space \(S\) then,
\(E_{1} \cup E_{2} \cup E_{3} \cup \ldots \ldots \ldots . . \cup\, E_{n}=S\)

9. Events Associated with “AND”

If two events \(E_{1}\) and \(E_{2}\) are associated with AND, then it means the intersection of elements that is common to both the events. The intersection symbol \((\cap)\) is used to represent AND in probability.

Thus, the event \(E_{1} \cap E_{2}\) denotes \(E_{1}\) and \(E_{2}\).

10. Event \(E_{1}\) but \(\operatorname{not} E_{2}\)

It represents the difference between both events. Event \(E_{1}\) but not \(E_{2}\) represents all the outcomes that are present in \(E_{1}\) but not in \(E_{2}\). Thus, the event \(E_{1}\) but not \(E_{2}\) is represented as \(E_{1}-E_{2}\).

Learn All Concepts on Probability

Solved Examples

Q.1. The sample space of an experiment is given as, \(S = \left\{{10,11,12,13,14,15,16,17} \right\}\) and the event, \(E\) is defined as the set of all the even numbers. Find the complementary event for \(E\)?
Ans: Given the sample space \(S = \left\{{10,11,12,13,14,15,16,17} \right\}\)
And the event, \(E\) is defined as all the even numbers, therefore
\(E(\text{ all even numbers})={10,12,14,16}\)
We know that, the complement of an event \(A\) is the event consisting of all sample space outcomes that do not correspond to the occurrence of \(A\).
Hence, \(E’\left( {{\rm{complementary}}\,{\rm{of}}\,E} \right) = \left\{ {11,\,13,\,15,\,17} \right\}\)

Q.2. Name the types of events obtained from the given below experiments,
(i) A coin is tossed for the \(5^{\text {th }}\) time and the event of getting a tail when the first four times, the result was a head.
(ii) \(S = \left\{{1,2,3,4,5} \right\}\) and \(E = \left\{4 \right\}\)
(iii) \(S = \left\{{1,2,3,4,5} \right\}\) and \(E = \left\{2,4 \right\}\)
(iv) \(S = \left\{{1,2,3,4,5} \right\}\), \({E_1} = \left\{{1,2} \right\}\) and \({E_2} = \left\{{3,4} \right\}\)
Ans: (i) Given that a coin is tossed for the \(5^{\text {th }}\) time and the event of getting a tail when the first four times, the result was a head. The likelihood of getting a tail will be \(0.5\) no matter how many times the coin is tossed. The event will thus be independent.
(ii) Given, \(S = \left\{{1,2,3,4,5} \right\}\) and \(E = \left\{4 \right\}\) We know that, any event consisting of a single point of the sample space is a simple event in probability. Hence, \(E={4}\) is a simple event.
(iii) Given, \(S = \left\{{1,2,3,4,5} \right\}\) and \(E = \left\{2,4 \right\}\)
We know that, every event that involves more than one single point of the sample space is referred to as a compound event.
\(E = \left\{2,4 \right\}\) is a compound event.
(iv) Given, \(S = \left\{{1,2,3,4,5} \right\},{E_1} = \left\{{1,2} \right\}\) and \({E_2} = \left\{{3,4} \right\}\)
Two events \(A\) and \(B\) of a sample space \(S\) are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. \(P\left(E_{1} \cap E_{2}\right)=0\)
Hence, \(E_{1}\) and \(E_{2}\) are mutually exclusive events.

Q.3. Create a sample space for the experiment consisting of a single die roll. Find the events that matches the phrases
1. Rolling of an even number
2. Rolling of a number greater than two
Ans: The outcomes could be labelled according to the number of dots on the top face of the die. Then the sample space is the set \(S = \left\{{1,2,3,4,5,6} \right\}\)
1. The outcomes that are even are \(2, 4\), and \(6\), so the event that corresponds to the phrase “an even number is rolled” is the set \(\left\{{2,4,6} \right\}\). It can be denoted as \(E = \left\{{2,4,6}\right\}\)
2. Similarly, the event that corresponds to the phrase “a number greater than two is rolled” is the set \(T = \left\{{3,4,5,6}\right\}\)

Q.4. Joshua and his friend bought movie tickets and ate a pizza. Can you tell us whether it is a dependent event or an independent event?
Ans: This is an independent event since buying movie tickets is unaffected by eating pizza.

Q.5. Three coins are tossed simultaneously.
\(P\) is the event of getting at least \(2\) heads.
\(Q\) is the event of getting no heads.
\(R\) is the event of getting heads on the second coin.
Which of the pairs is mutually exclusive?
Ans: \(n(S)=2 \times 2 \times 2=8\)
\(n(P)=H H T, H T H, T H H, H H H=4\)
\(n(Q)=T T T=1\)
\(n(R)=T H T, H H H, H H T, T H H=4\)
So \((Q, R)\) and \((P, Q)\) are mutually exclusive as they have nothing in their intersection.

Summary

The essential definitions of random experiment, outcome, and sample space have been covered in this article. Then we went over exactly what an event is and what the various types of events are.

We learned about impossible and certain events, simple events, compound events, independent and dependent events, mutually exclusive events, exhaustive events, complementary events, events linked with OR & AND, event \({E_1}\) but not \({E_2}\) and solved examples.

Frequently Asked Questions

We have provided some frequently asked questions on types of events here:

Q.1. How do you tell if an event is independent or dependent?
Ans: If the occurrence of one event does not affect the probability of the occurrence of the other, two events \(A\) and \(B\) are said to be independent. The two occurrences are also said to be dependent if the occurrence of one event affects the probability of the occurrence of the other.

Q.2. Are dependent events mutually exclusive events?
Ans: Two mutually exclusive events are neither necessarily independent nor dependent.

Q.3. What is the probability of an impossible event and a sure event?
Ans: The probability of a sure event is always \(1\), while the probability of an impossible event is always \(0\).

Q.4. What is event and examples?
Ans: The set of outcomes that we get from an experiment is an event. So an example would be when we toss a coin. The result of this means the coin can either land on the ‘heads’ side or ‘tails’.

Q.5. Write the difference between a sample space and an event?
Ans: A collection of possible outcomes of a random experiment is defined as a sample space, while an event is the subset of sample space. For example, when a die is rolled, the sample space will be \(\left\{ {1,~2,~3,~4,~5,~6} \right\}\) and the event of getting an even number will be \(\left\{ {~2,~4,~6} \right\}\).

We hope you find this detailed article on types of events helpful. If you have any doubts or queries regarding this topic, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!