we already know what is probability in earlier classes. In this post we will discuss concepts which are relevant for Grade 11 Mathematics.
Let us revise the basics of probability first.
Probability Basics
Probability basically tells the chance of happening any event.
The basic example used to explain the concept of probability is “Rolling of Dice”
With the help of probability we can find the chance of getting number 2 on rolling the dice.
Numbers on dice is {1, 2, 3, 4, 5, 6}
Total Number of Possible Outcomes ⟹ 6
Outcome we need = [Number 2]
Favorable Outcome ⟹ 1
Hence the chance of getting number 1 after rolling a dice is 1/6.
Note that probability does not provide you with the prediction of happening a event. It just gives you an idea about the chance or likelihood of event occurrence.
Now this is just the basic concept of Probability. As we move to higher grade mathematics, the problems get more complex and need advanced math concepts to solve.
In this post we will try to understand some important concepts of probability which will help to solve problems related to class 11 Math NCERT and CBSE syllabus.
Sample Space in Probability
Sample space in probability is a set which lists down all the possible outcome of any event.
For Example:
(1) Rolling a dice
The possible outcomes are, S ⟹{1, 2, 3, 4, 5, 6}
These set of outcomes are known as Sample Space of the event and each element in the set is called Sample Point
(2) Tossing two coins at once
The possible outcomes are, S ⟹ {HH, HT, TH, TT }
You can see that there are 4 possible outcomes in sample space
(3) Tossing coin repeatedly until head comes up
Possible Outcomes are, S ⟹ { H, TH, TTH, TTTH, TTTTH, . . . . . . }
Here the sample space is an infinite series.
Let us now understand the concept of event in this chapter.
Event in Probability
Event is the set of favorable outcomes in any given experiment.
Event is also subset of Sample Space
Example
(1) Rolling a Dice
Event A : Number less than 3
Sample Size S = {1, 2, 3, 4, 5, 6}
Event A = {1, 2 }
Here Event A is a set of all the desired outcomes
Event B : Number greater than 2 but less than 5
Event B = { 3, 4}
Event C: Number greater than 4
Event C : {5,6}
(2) Tossing two coin simultaneously
Sample Size S = {HH, HT, TH, TT}
Event A: Exactly one head appeared
Event A : { HH, HT, TH}
Event B : At most one head appeared
Event B : {HT, TH, TT}
(3) Tossing three coin simultaneously
Sample Size S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Event A: Exactly one Tail Appeared
Event A : {HHT, HTH, THH}
Event B: Atleast one Tail Appeared
Event B: { HHT, HTH, HTT, THH, THT, TTH, TTT}
Now you have understood the basic concept of event in probability. Let us now understood the types and properties of events.
Types of Events
(01) Impossible Events
Any event which is impossible to occur in a given experiment is termed as impossible event.
It is denoted by \phi
Example
Sample Size of Rolling a Dice = {1, 2, 3, 4, 5, 6}
Event A = Number 7 appeared
Understand that the event A is impossible to occur because there is no number 7 in the dice. Hence we represent Even A as impossible event.
Event A = \phi
(02) Sure Event
Any event which consist of the whole sample space is termed Sure Event
Example:
A Pack of 52 cards, where 26 cards are red and remaining 26 cards are black in color
Sample Size S = {B1, B2, B3 . . . . .B26, R1, R2, R3 . . . . .R26}
Event A : Select one card which is either Red or Black
Event A : {B1, B2, B3 . . . . .B26, R1, R2, R3 . . . . .R26}
You can see in the above example that event set A is equal to the Sample size. Hence event A is a sure event.
(03) Simple Event and Compound Event
I event contain exactly one element than it is known as Simple Event.
And if event has more than one sample point than it is known as compound event
Example
Consider tossing of three coins simultaneously
{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Event A: All heads
Event A : { HHH}
Since Event A contains only one element it is considered simple event
Event B : Atleast one Head
Event B: { HHH, HHT, HTH, HTT, THH, THT, TTH}
Event B contains multiple heads, hence it is a compound event
Algebra of Events in Probability
In this topic we will understand some basic algebra operations of events. To understand the concept fully, you should have basic idea of set theory concept.
(01) Complementary Event
Complementary Event basically contains element which is left out by the event.
For Event A, the complementary event is represented by A’
Example:
let Sample Size S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Event A = {1, 2, 3, 4, 5, 6}
The complement of event A contain left out elements of A
A’ = S – A
A’ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 2, 3, 4, 5, 6}
A’ = {7, 8, 9, 10}
(02) Union of Events “A or B” (A U B)
Union of events contain element present in A and B
Let Sample Size = {11, 12, 13, 14, 15, 16, 17}
Event A = {11, 12, 13}
Event B = {15, 16, 17}
so A U B ⟹ {11, 12, 13, 15, 16, 17}
(03) Intersection of Event “A & B” ( A\cap B )
Intersection of event basically takes the common element present between A and B
Let Sample Size = { 101, 102, 103, 104, 105, 106, 107}
Event A = {101, 102, 103}
Event B = {103, 104, 105, 107}
( A\cap B ) = {103}
Note that we have taken common element between Event A and Event B
(04) A but not B [ “A – B”]
In this operation we will remove common element of A & B from Event A.
Example
Sample Size S = [50, 51, 52, 53, 54, 55, 56]
Event A = [50, 51, 52, 53, 54]
Event B = [53, 54, 55, 56]
A – B = [50, 51, 52]
(05) Mutually Exclusive Events
Two Events A & B are Mutually Exclusive if there is no common element between the two
For Example
Sample Size S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Event A = {1, 2, 3, 4, 5}
Event B = {6, 7, 8, 9}
From the above two events A & B, you can observe that there is no common element between the two. Hence they are said to be Mutually Exclusive events.
We can also write that:
A\cap B=\phi
A intersection B is nil as there is no common element.
Set A and B are also called disjoint set
(06) Exhaustive events
All the events are said to be exhaustive events if their union results in sample size
Foe Example
Sample Size = {11, 12, 13, 14, 15, 16 , 17}
Event A = {11, 12, 13}
Event B = {14, 15}
Event C = {16, 17}
Now let us calculate the union of all the events
A U B U C = {11, 12, 13, 14, 15, 16 , 17}
you can see that the above union set is equal to the sample size
Hence events A, B, C can be termed as exhaustive events
Questions Related to Sample Size and Events
(01) An experiment involves rolling a pair of dice and recording the number that come up.
Some of the events for this experiment are:
Event A: Sum is greater than 8
Event B: 2 occurs on either die
Find
(a) the sample size of the experiment
(b) Find A\cap B
(c) Check if Event A and B are mutually exclusive
Solution
(a) Sample Space for the experiment is given by:
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)}
(b) Event A : Sum is greater than 8
Event A : {(3,6), (4,5), (5,4), (6,3), (4,6), (6,4), (5,5), (6,5), (5,6), (6,6)}
Event B : 2 occurs on either die
Event B : { (2,1), (2,2), (2,3), (2,4), (2,5),(2,6), (1,2), (3,2), (4,2), (5,2), (6,2) }
You can see that there is no common element between A & B, hence the sets are mutually exclusive
A\cap B=\phi
(02) From a group of 2 boys and 3 girls, two children are selected at random.
Describe the following events:
(a) Sample Size
(b) Event A : Both selected children are girls
(c) Event B : the selected group consist of one boy and one girl
Solution
Let the two boys and three girls are : B1, B2, G1, G2, G3
(a) Sample Size
S = [B1B2, B1G1,B1G2, B1G3, B2G1, B2G2, B2G3, G1G2, G1G3, G2G3]
(b) Event A: Both selected children are girls
Event A = [G1G2, G1G3, G2G3]
(c) Event B : One boy and one girl
Event B = [B1G1,B1G2, B1G3, B2G1, B2G2, B2G3]
(03) In a single throw of dice, describe the following events
(i) A: Getting number less than 7
(ii) B: Getting number greater than 7
(iii) C: Getting multiple of 3
(iv) D: Getting number less than 4
Also find AUB, A\cap B,\ B\cap C
Solution
let us first define the sample size of the above experiment
Sample Size S={1, 2, 3, 4, 5, 6}
(i) A : Getting number less than 7
A : {1, 2, 3, 4, 5, 6}
(ii) B: Getting number greater than 7
B : \phi
(iii) C: getting multiple of 3
C : {3, 6}
(iv) D : number less than 4
D : {1, 2, 3}
(v) A U B = { 1, 2, 3, 4, 5, 6}
A\cap B = \phi
B\cap C = \phi