Types of Events in Probability

In our life, we often use statements like, “it might rain today” or “I might not turn out today” which indicate the probability or certainty of something happening. In maths, probability helps in determining the exact event to occur. Events in probability take place as a result of one or more outcomes of random experiments. The toss of a coin, throw of a dice or lottery draws are all instances of random events. 

The topic “Types of Events” is covered in the Unit 6 Chapter 16 i.e, Probability of CBSE Class 11 Mathematics. The whole unit 6 i.e, Statistics and Probability will carry a weightage of 21 periods and students can expect around 10-12 marks from this unit in the CBSE class 11 examination. 

Define Events in Probability?

A set of outcomes of an experiment is called a probability. In other words, an event in probability is defined as the subset of the respective sample space. Therefore, the outcomes of a random experiment of tossing an unbiased coin are “head” and “tail”, which are the events related to this experiment. Likewise, the results of a random experiment of throwing an unbiased die from a box are, 1, 2, 3, 4,...., 6 where each outcome is an event connected with this experiment. 

The video below explains this:

Types of Events in Probability Detailed Video Explanation:

Sample space

The individual space of an experiment or the entire possible set of outcomes of a random experiment is called the sample space.The conceivability of an event is known as probability.

0 and 1 is the chance that occurs has a probability of occurrence of any event.

The probability for the tossing of three coins simultaneously is given by:

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

If we want to find only two heads outcomes,then the possibility can be given below.

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Hence, the sample space has an event as the subset, i.e., E is a subset of S.

Sample space has lots of events associated with the given space.If any event should occur,then the outcome of the demonstration must be an element of the set of event E.

The Probability of Occurrence of an Event

The probability of occurrence of an event is known as the number of favourable outcomes to the total number of outcomes.

 The probability that an event will occur is given below:

P(E) = Number of Favourable Outcomes/ Total Number of Outcomes

Types of Events in Probability:

Some of the important probability events are:

• Impossible and Sure Events
• Simple Events
• Compound Events
• Independent and Dependent Events
• Mutually Exclusive Events
• Exhaustive Events
• Complementary Events
• Events Associated with “OR”
• Events Associated with “AND”
• Event E1 but not E2

Impossible and Sure Events

If an event has 0 number of occurrences in total probability then it is called an impossible event. If an event has 1 number of occurrences in total probability then it is called a sure event. In other words, the set that will be always empty ? is an impossible event and the sample space S is a sure event.

Simple Events

In probability any event consisting of a single point of the sample space is known as a simple event. For example:

If S = {56 , 78 , 96 , 54 , 89} and E = {78}

 Then E is a simple event.

Compound Events

The compound event has more than one single point of the sample space then the event is known as compound event.For example take the same example as before if S = {56 ,78 ,96 ,54 ,89}, E1 = {56 ,54 }, E2 = {78 ,56 ,89 } then, E1 and E2 indicates two compound events.

Independent Events and Dependent Events

The occurrence of the independent event does not affect the occurrence of the other event.If the occurrence of the event is affected by another event then that event is known as dependent events.

Mutually Exclusive Events

The Mutually exclusive event excludes the occurrence of another event.For example two events did not have any common point. 

For example, if S = {5 , 6 , 7 , 8 , 9 , 10} and F1, F2 are two events such that F1 consists of numbers less than 7 and F2 consists of numbers greater than 8.

So, F1 = {5,6} and F2 = {9,10} .

Then, F1 and F2 are mutually exclusive.

Exhaustive Events

In probability an event that covers all the probability space is called exhaustive.In this event the entire sample space is consumed all together.

Complementary Events

For any event F1 there has been another event F1‘ which indicates the remaining elements of the sample space S.

F1 = S − F1‘

For example if we roll a dice then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }.

 If event F1 indicates all the outcomes which are bigger than 4, then F1 = {5, 6} and F1‘ = {1, 2, 3, 4}.

Thus F1‘ is the complement of the event F1.

Likewise, the complement of F1, F2, F3……….Fn will be known as F1‘, F2‘, F3‘……….Fn‘

Events Connected with “OR”

The union symbol (∪) is used to indicates OR in probability.Any two events that were associated with OR that represents both the two events.

Thus, the event F1U F2 denotes F1 OR F2.

If have the events that are mutually exclusive like F1, F2, F3 ………Fn connected with sample space S then,

F1 U F2 U F3U ………Fn = S

Events Connected with “AND”

If two events that are associated with AND then it denotes the intersection of elements which is common to both the events. The symbol (∩) is known as intersection symbol that is used to represent AND in probability.

Thus, the event F1 ∩ F2 denotes E1 and F2.

Event F1 but not F2

This event represents the difference between both the events. Event F1 but not F2 represents all the outcomes which are available in F1 but not in F2. 

Hence, the event F1 but not F2 is represented as

F1, F2 = F1 – F2

The video below explains this:

Independent Events Detailed Video Explanation:

Formulas in Probability

Classical Probability Formula

It is based on the fact that all outcomes are equally possible.

Total number of outcomes in E

P(E)= ________________________________________________

Total number of outcomes in the sample space

For better understanding of the topic, refer to this video: 

Solved Examples

Ques: In the game of snakes and ladders, a fair die is thrown. If event F1 declares all the events of getting a natural number less than 4, event F2 consists of all the events of getting an even number and F3 indicates all the events of getting an odd number. List the sets representing the following:

i)F1 or F2 or F3

ii)F1 and F2 and F3

iii)F1 but not F3

Ans:

The sample space is given as S = {1 , 2 , 3 , 4 , 5 , 6}

F1 = {1,2,3}

F2 = {2,4,6}

F3 = {1,3,5}

i)F1 or F2 or F3= F1 F2 F3= {1, 2, 3, 4, 5, 6}

ii)F1 and F2 and F3 = F1 F2 F3 = ∅

iii)F1 but not F3 = {2}

Ques: The table below shows students distribution per grade in a school.

If a student is selected at random from this school then what will be the probability that this student is in grade 3?

Ans: Let event E be "student from grade 3". 

Hence

Frequency for grade 3

P(E)=

_______________________________________

Total frequencies

= 40 / 250 = 0.16.

Ques:Two dice are rolled, find the probability that the sum is

1. a) equal to 1
2. b) equal to 4
3. c) less than 13

Ans: 

1. a) The sample space S of two dice is shown below.

S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

Let E be the event "sum equal to 1". There are 0 outcomes which correspond to a sum equal to 1.

 Hence

P(E) = n(E) / n(S) = 0 / 36 = 0

1. 
   b) 3 possible outcomes give a sum equal to 4: 

E = {(1,3),(2,2),(3,1)}

 Hence,

P(E) = n(E) / n(S) = 3 / 36 = 1 / 12

13. 
    c) All possible outcomes, E = S, give a sum less than 13.

 Hence.

P(E) = n(E) / n(S) = 36 / 36 = 1

Ques: A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white?

Ans:

First we should construct a table of frequencies that gives the marbles color distributions 

Now we use the empirical formula of the probability

P(E) = Frequency for white color / Total frequencies in the above table

= 10 / 20 = 1 / 2

Topics that are Related to Probability Events

Sample Questions

Ques: What is an Event in Probability Example?

Ans: The set of outcomes from an experiment is known as an Event.for example if we toss a coin then the outcome of that experiment is landing in even “head” or “tail” this is said to be an event conducted with the experiment.

Ques: What are the 3 types of events?

Ans: Events can be defined on the basis of their size, type and context.There are three main categories which events were considered under. These events are private, corporate and charity.

Ques: What is the Probability of an Impossible Event and a Sure Event?

Ans: In probability the sure event is always 1 and the probability of an impossible event is always 0.

Ques: What is an Example of an Impossible Event?

Ans: The probability of getting more than 6 when a die is rolled.

Ques: A bag contains 27 balls within which 10 are red, 2 are green and the rest are white. Annie takes out a ball from the bag at random. What is the probability that she takes 

(i) a white ball

(ii) a ball that is red or green

Ans: (i) number of white balls

27 - (10 + 2) = 15

P (white ball) = Number of white balls/ Total number of balls = 15/27 = 5/9

(ii) P (red ball or green ball) = P(red) + P(green)

= 10/ 27 + 2/27 = 12/ 27= 4/9.

Ques: Explain combined events.
Ans: There are many times when two or more events occur together. Some of the common examples are:

• Tossing two coins together

There are four equally likely outcomes: {HH, HT, TH, TT}

• Tossing three coins together

There are eight equally likely outcomes: {HHH, HTT, THH, TTH, HHT, HTH, THT, TTT}

Ex. P (3 heads or 3 tails) =? + ? = 2/8 = ¼

Ques: If A and B are mutually exclusive events of an experiment. If P(not A) = 0. 65, P(A ∪ B) = 0. 65, and P(B) = P, then find out the value of P.

Ans: Given:

P(not A) = 0. 65

⇒ 1 - P(A) = 0. 65

⇒ P(A) = 0. 35

P(A ∪ B) = 0. 65, and P(B) = P

As A and B are mutually exclusive events, therefore

P(A ∪ B) = P(A) + P(B)

⇒ 0.65 = 0. 35 + P

⇒ P = 0. 30

Ques: Given two mutually exclusive events A and B such that P(A) = ½ and P(B) = ? , find P(A or B)

Ans: Given A and B are two mutually exclusive events

And, P(A) = ½ P(B) = ? 

We have to find P(A’ or B’)

P(A or B) = P(A ∪ B)

By the definition of the mutually exclusive events, it is already known that,

P(A ∪ B) = P(A) + P(B) 

Therefore, P(A ∪ B) = P(A) + P(B) 

= ½ + ? = ? 

CBSE Class 11 Mathematics: Important Highlights

• The CBSE Class 11 Physics Question paper will consist of 29 questions in total where all the questions are compulsory.
• Questions 1 to 4 in Section A will be very short answer type questions and will carry 1 mark each.
• Questions 5 to 12 in Section B will be short answer type questions and will carry 2 marks each.
• In Section C, question number 13 to 23 will be long answer type questions which will carry 4 marks each.
• In Section D, questions 24 to 29 will be long answer type II questions and will carry 5 marks each.
• There will be no overall choice.
• However, an internal choice will be provided in one question of 2 marks, one question of 3 marks, and all three questions of 5 marks each. 
• The whole paper will be based on 5 categories: Remembering (Knowledge-based), Understanding (comprehension), Application based, Higher order thinking skills and Evaluation.