The document provides an overview of concepts related to probability and statistics, including definitions of random experiments, sample spaces, events, and types of events such as mutually exclusive, independent, and compound events. It explains the methods for calculating probabilities and gives examples of various experiments, including die rolls and card draws. Further, it touches on permutations and combinations, outlining their definitions and calculating methods.

1. By
Tushar Bhatt
4/12/2019Atmiya University - Rajkot 1

2. Content
 Introduction
 Random Experiments
 Sample space
 Events and their probabilities
 Random variable probability distribution
 t-test, Paired t – test, F – test (ANOVA)
 Comparison of results of above tests
4/12/2019Atmiya University - Rajkot 2

3.  Introduction
4/12/2019Atmiya University - Rajkot 3
 Def : Experiment/ Random experiment
An activity or phenomenon that is under consideration is called an
experiment .
 Def : Trial
A single performance of an experiment is briefly called a trial.
 Def : Outcome or Sample points
The experiment can produce a variety of results called outcomes or
sample points.

4.  Introduction
4/12/2019Atmiya University - Rajkot 4
 Def : Sample space
The set of all possible outcomes of an experiment is called sample
space and it is denoted by S .
 For example :
1. A random experiment of tossing a coin then S = { H, T }
2. A random experiment of throwing a die then S = { 1,2,3,4,5,6 }
3. If two coins are tossed simultaneously then
S = { (H, H), (T, T), (H, T), (T, H) }

5.  Introduction
4/12/2019Atmiya University - Rajkot 5
 Def : Event
Any subset of sample space is called event.
 For example :
1. A random experiment of throwing a die then S = { 1,2,3,4,5,6 }
let A be an event that an odd number appears on the die then
A = { 1, 3, 5 }

6.  Introduction
4/12/2019Atmiya University - Rajkot 6
 Def : Favourable outcomes
If an event A is defined on sample space S then the sample points
which are included in A are called favourable outcomes for the
event A.
 For example :
1. A random experiment of throwing a die then S = { 1,2,3,4,5,6 }
let A = { 1,2,3,4 } in which 1,2,3,4 are favourable outcomes.

7.  Types of event
4/12/2019Atmiya University - Rajkot 7
 Def : Exhaustive events
The total number of all possible outcomes of an experiment is
called exhaustive events.
 For example :
1. In tossing a coin there are two exhaustive events (a) Head and
(b) Tail.

8.  Types of event
4/12/2019Atmiya University - Rajkot 8
 Def : Mutually exclusive events
-Events are said to be mutually exclusive or disjoint or
incompatible, if one and only one of them can take place at a time .
- If then the events A and B are said to be mutually
exclusive events.
 For example :
1. In tossing a coin the events (a) Head and (b) Tail are mutually
exclusive.
A B  

9.  Types of event
4/12/2019Atmiya University - Rajkot 9
 Def : Equally likely events
-Events are said to be equally likely , if one event can not perform
without help of other event.
 For example :
1. In tossing a coin the events (a) Head or(b) Tail are equally
likely events.

10.  Types of event
4/12/2019Atmiya University - Rajkot 10
 Def : Independent events
-Two or more events are said to be independent, if in performance
of one event does not effect on performance of other event.
- Otherwise they are said to be dependent.
 For example :
1. In tossing a coin, the result of the first toss does not effect on
result of the second toss.

11.  Types of event
4/12/2019Atmiya University - Rajkot 11
 Def : Certain events
-The event which is sure to occur, is called certain event.
- Sample space S is certain event.
 Def : Impossible events
- The event which is impossible to occur at all, is called impossible
event and it is denoted by 
 For example :
1. In tossing a die, the event that number 7 will occur is
impossible event.

12.  Types of event
4/12/2019Atmiya University - Rajkot 12
 Def : Favourable events
- All those events which result in the occurrence of the event,
under consideration is called favourable event.
 For example :
1. In throwing two die, the favourable events of getting the sum 5
is (1, 4), (4, 1), (2, 3), (3, 2).

13.  Types of event
4/12/2019Atmiya University - Rajkot 13
 Def : Compound events
- When two or more events occur in connection with each other,
their simultaneous occurrence, is called compound event .
 Def : Probability
-Let ‘n’ be the number of equally likely, mutually exclusive and
exhaustive outcomes of a random experiment.
- Let ‘m’ be the number of outcomes which are favourable to the
occurrence of event A, then the probability of event A occurring
denoted by P(A) is given by
.
( )
.
No of outcomes favourable to A m
P A
No of exhaustive outcomes n
 

14.  Probability
4/12/2019Atmiya University - Rajkot 14
 There are n-m outcomes in which event A will not happen, the
probability , that event A will not occur is denoted by P(A’),
defined as
'
'
'
'
.
( ) 1 1 ( )
.
( ) 1 ( )
( ) ( ) 1
0 ( ), ( ) 1
unfavourable No of outcomes n m m
P A P A
No of exhaustive outcomes n n
P A P A
P A P A
Note that P A P A

     
  
  
 

15.  Probability
4/12/2019Atmiya University - Rajkot 15
 Note :
1. If , A is certain event.
2. If , A is impossible event.
3. If A and B are two events then the probability of occurrence of
at least one of the two events is given by
( ) 1P A 
( ) 0P A 
( ) ( ) ( ) ( )P A B P A P B P A B    

16.  Probability
4/12/2019Atmiya University - Rajkot 16
 Ex – 1 : Draw a number at random from the integer 1 through
10. What is the probability that a prime is drawn.
Solu : Here S = { 1,2,3,4,5,6,7,8,9,10 }
Total number of possible outcomes = 10
A = An event , draw a prime no’ from 1 to 10 = { 2,3,5,7 }
Favourable outcomes = 4
. 4 2
( )
10 5
No of favourable outcomes
P A
Exhaustive outcomes
  

17.  Probability
4/12/2019Atmiya University - Rajkot 17
 Ex – 2 : A box contains 4 red balls, 3 green balls and 5 white
balls. If a single ball is drawn, what is the probability that it is
green ?
Solu : Here S = { 4-red balls, 3-greeen balls, 5-white balls }
Total number of possible outcomes = 12
A = An event , when green ball is drawn from box
Favourable outcomes = 3
. 3 1
( )
12 4
No of favourable outcomes
P A
Exhaustive outcomes
  

18.  Probability
4/12/2019Atmiya University - Rajkot 18
 Ex – 3 : In rolling a fair die, what is the probability of A,
obtaining at least 5 ? And the probability of B, obtaining even
numbers ?
Solu : Here S = { 1,2,3,4,5,6 }
- Total number of possible outcomes = 6
- A = An event, obtaining at least 5 = { 5, 6 }
- Favourable outcomes = 2
- B = An event, obtaining even numbers = { 2,4,6 }
- Favourable outcomes = 3
2 1 3 1
( ) ; ( )
6 3 6 2
m m
P A P B
n n
     

19.  Permutations
4/12/2019Atmiya University - Rajkot 19
 A permutations is an arrangement of a number of objects in a
definite order.
 For example :
 There are 3-letters A , B and C arrange in 6-different ways = 3!
 If we have n-different objects to arrange then the total number
of arrangements = n!
 The number of arrangements of n- different objects taking r at
a time is
 The number of arrangement of n-different objects taking r at a
time with repetition is
!
( )
( )!
n
r
n
P Without repetition
n r


r
n

20.  Permutations
4/12/2019Atmiya University - Rajkot 20
 Ex-4 : How many arrangement can be made of the letters A, B,
C, D, E taking two letters at a time , if no latter can be repeated?
 Solu : Here n=5 and r = 2
5
2
!
( )
( )!
5! 5*4*3*2*1
20 .
(5 2)! 3*2*1
n
r
n
P Without repetition
n r
P total no of arrangement


   


21.  Permutations
4/12/2019Atmiya University - Rajkot 21
 Ex-4 : How many arrangement can be made of the letters A, B,
C, D, E taking two letters at a time , if repetition is under
consideration?
 Solu : Here n=5 and r = 2
5 2
2
( )
5 25 .
n r
rP n With repetition
P total no of arrangement

  

22.  Combinations :
4/12/2019Atmiya University - Rajkot 22
 A Combination is a selection of a number of objects in any
order.
 Here selection is important not an order(arrangement).
 For example :
 AB and BA represent the same selection however AB and BA
represent different arrangements.
 It is denoted by
 it gives the number of ways of choosing r- objects from n-
different objects.
!
r!( )!
n
r
n n
C
r n r
 
  
 n
r
 
 
 

23.  Combinations :
4/12/2019Atmiya University - Rajkot 23
 Ex-1 : Ten people take part in a chess competition. How many
games will be played if every person must play each of the others
?
 Solu : We have 10 people.
We want to choose 2 ( as 2 people play in each game )
Thus n = 10 and r = 2.
Number of games are
10
2
10 10!
45
2 2!(10 2)!
C
 
   
 

24.  Note :
4/12/2019Atmiya University - Rajkot 24
 If A and B are mutually exclusive events then occurrence of
either A or B is given by ( ) ( ) ( )P A B P A P B  
 ( ) ( ) ( ) ( ) ( ) (B ) ( )P A B C P A P B P C P A B P C P A B          

P( ) 1 ( )A B C P A B C     

P( ) ( ) ( )A B P B P A B   

25. 4/12/2019Atmiya University - Rajkot 25

26. 4/12/2019Atmiya University - Rajkot 26

27. 4/12/2019Atmiya University - Rajkot 27

28. 4/12/2019Atmiya University - Rajkot 28

29. 4/12/2019Atmiya University - Rajkot 29

30. 4/12/2019Atmiya University - Rajkot 30

31. 4/12/2019Atmiya University - Rajkot 31

32. 4/12/2019Atmiya University - Rajkot 32

33. 4/12/2019Atmiya University - Rajkot 33

34. 4/12/2019Atmiya University - Rajkot 34
Comparison of parametric tests :

35. 4/12/2019Atmiya University - Rajkot 35