Introduction to
probability
MAHESH MT
ASSISTANT PROFESSOR OF COMMERCE
KNM GOVT. ARTS AND SCIENCE COLLEGE

 Introduction to Probability
 Certain Possible and Impossible Events
 Sample space
 The compliment of an event
 Mutually exclusive events
 Addition and multiplication rules for probability.
 Independent and dependent events.
 Combinatorial probability
 Conditional probability
 Bayes theorem/ Law of total probability
 Central limit theorem
CONTENTS

 “It is likely to rain today.”
 “I have a fair chance of getting admission.”
 “There is an even chance that in tossing a coin, the head may
come up.”
 In each case, we are not certain of the outcome, but we wish
to assess the chances of our predictions coming true.
 If we assign numerical value, the statement would become
more precise. The theory of probability provides a numerical
measure of the element of uncertainty.
 Probability refers to the chance of happening or not
happening an event.
Introduction to Probability

 Experiment – It denotes to describe an act which can be
repeated under some given conditions.
 Random experiment: An experiment has two or more
outcomes which may vary in an unpredictable manner
from trial to trial when conducted under uniform
conditions. It is called ‘random experiment.’
 Trial : Performing a random experiment. (Tossing a coin)
 Outcome: This is the result of a random experiment.( T)
 Event: An outcome or a combination of outcomes of a
random experiment .( HH)
Terms used in probability

 The theory of probability has its origin in the ‘games of
chance ’’ related to gambling such as tossing a coin a,
drawing cards from a pack of cards etc.
 Jerame Cardon (1501-76), an Italian mathematician, was
the first person who wrote a book named ‘Book on
games of chance’.
 Starting with the games of chance, probability has
become one of the important statistical tool today.
 A knowledge of probability is required to interpret
statistical results as many statistical procedures involve
conclusions based on samples.
origin

 sample point/ sample element/elementary outcome:
Every indecomposable outcome of a random
experiment is called a sample point of that random
experiment.
Eg: tossing a coin and getting Head is a sample point.
Sample space: It is the set which contains all the sample
points of that random experiment. It is totality of all the
elementary outcome of that experiment.
A sample space may be discrete or continuous in nature.
Eg: tossing a coin twice, the sample space is HH, HT,TH and
TT denotes as S={HH, HT,TH, TT}
Terms used in probability

 Certain event : An event whose occurrence is inevitable is
called certain event.
Eg: Getting a blue ball from a bag containing all blue balls.
Impossible event: - in the above example, getting a black ball
from the set of blue balls is an impossible event. An empty set
can be denotes as ø. Ø= null set.
Uncertain event:- it is the event which cannot predict its
happening. The happening of that event is neither sure nor
impossible.
Eg: Getting a black ball from a bag containing white and black
balls. It is uncertain.
Terms used in probability

 Equally likely events:- Two or more events are said to be equally
likely or equally probable if each one of them has equal chance
of occurring.
 Eg: while tossing a coin, getting a Head or tail has equal chance
and hence it is equally likely.
 Mutually exclusive events: Two or more events are said to be
mutually exclusive if the happening of any one of them
excludes the happening of all others in the same experiment.
 Eg: when a coin is tossed, the occurrence of Head completely
excludes the occurrence of tail.
Terms used in probability

 Exhaustive events: A group of events is said to be exhaustive
event, when it includes all possible outcomes of the random
experiment under consideration.
 Eg: while tossing a single coin, the outcome will be head
/tail. Head and Tail together will form an exhaustive event
because one of them will occur when a coin is tossed.
 Mutually exclusive and exhaustive events/ complementary
events: A set of events said so, if one of them must and only
one can occur.
 Eg: while tossing two coins, getting no head or getting one
head or getting two heads are mutually exclusive and
exhaustive.
Terms used in probability

 Compound events : refers to the joint occurrence of two or
more events. Thus compound events imply the simultaneous
occurrence of two or more simple events.
 Independent events: Two or more events are said to be
independent if the occurrence of one of them in no way
effects the occurrence of other event.
 Eg: tossing a coin twice. The result of second tossing is not
effected by the result of first tossing.
 Dependent events: The happening of one event in any one
trial affects the probability of other events in other trials.
 Eg: Drawing a ‘King’ first from a packet of cards, then draws
another king without replacing the first king in the packet of
cards. The second event is dependent event.
Terms used in probability

 Meaning :
 The number of occasions that a particular event is likely to
occur in a large population of events. A particular event can
be expressed positively where the event is likely to happen
and negatively when the event is not likely to happen. Ranges
from 0 to 1.
 Definition :if the probability can be calculated even before the
actual happening of the event, ie even before conducting the
experiment is mathematical probability.
 if the probability can be determined only after the actual
happening of the event, ie after conducting the experiment is
statistical probability.
Meaning and Definition of
probability

1. objective Probability
 a) Classical /priori / mathematical probability
( the outcomes of a random experiment are equally likely)
b) Empirical / Posteriory / Statistical Probability
( used when it is not possible to make a simple enumeration
of a case which can be considered equally likely)
2. Subjective / Personalistic Probability
( Probability as a measure of personal confidence in a
particular proposition)
3. Modern / Axiomatic Approach to Probability
(combines both subjective and objective concepts )
CONCEPTS USED IN PROBABILITY

 The probability is the ratio of the number of favourable cases to the
total number of equally likely cases – Laplace (French mathematician)
 This is also referred as ‘classical probability’ and ‘priori probability’.
 Probability of occurrence P(A)=success=
𝑚
𝑛
=
𝑛𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
 Probability of non occurrence Q =failure =P(not A) =
𝑛−𝑎
𝑛
 This formula may not be applied under the following situations
1. if it is not possible to enumerate all possible outcomes.
2. If the sample points are not mutually independent.
3. If the total number of items are infinite.
4. If each and every outcome is not equally likely.
Note: Priori means “from cause to effect’’.
1. Mathematical probability

 Q. From a bag containing 10 black and 20 red balls , a ball is
drawn at random.
 1. What is the probability that it is a black ball ?
 2. What is the probability that it is not a black ball ?
 Solution 1.
 1. total number of cases =10+20=30
 2.number of black balls =10 (favourable cases)
 Probability of getting a black ball q= m/n= 10/30=1/3
 Solution 2
 Probability of not getting a black ball= n-m/n
 =30-10/30=20/30=2/3
 Then p + q = 1/3 + 2/3 =1
Example

 If the probability of an event can be determined only after the actual
happening of an event.
 This approach was initiated by the mathematician VON MISES
 used when it is not possible to make a simple enumeration
of a case which can be considered equally likely.
 Probability of an event can here be defined as the relative frequency
with which it occurs on an indefinitely large number of trials .
 if an event occurs ‘m’ times out of ‘n’ times ,its relative frequency is
m/n
 in the limiting case , when ‘n’ becomes sufficiently large it
corresponds to a number which is called the probability of an event.
 Here P(A)= lim
𝑛→∞
(𝑚/𝑛)
 When this experiment is carried out a large number of times, the
expected outcome approached to 0.5
2. Statistical probability/ posteriori

Subjective theorists regard Probability as a measure of
personal confidence in a particular proposition.
Eg: A belief that a particular students of a college will
secure rank in the university examination . Here the
subjectivist can assign a weight between zero and one
to an even according to his belief.
It may vary from person to person , hence it is
personalistic probability
EG: I BELIEVE THAT YOU ALL WILL PASS THIS EXAM
Subjective approach

 A,B,C are used to denote sets ,
 let A be the first set and B be another set.
 Where A =(3,4,6,7) and B = (1,2,3,4,5)
 Thus A and B or A U B (A Union B)=(1,2,3,4,5,6,7)
 The event which consists of all points that are in A or B or in both
is called the union of A and B. It is written as A U B
INTERSECTION : The event which includes all points which are
common in both sets is called 𝐴 ∩ 𝐵 ( A Intersection B)
Here 3 and 4 are in both sets
SET THEORY

• This approach is developed by Russian Mathematician A.N Kolmogrove
in the year 1933.
• It combines both subjective and objective probability
• When this approach is followed , no precise definition of probability is
given ,rather we give certain axioms or postulates on which probability
calculations are based
• 1. The Probability of an event ranges from zero to one.
If the event cannot take place, the probability will be zero
If the event take place, the probability shall be one
symbolically 0≤P(A)≤1
• 2. The sum of all probabilities of all sample events constituting that
sample space is equals to 1 ie, P(S)=1
• 3. The probability of compound event is the sum of the probabilities of
sample events. If A and B are mutually exclusive, then Probability of
either A or B is denoted by P(A U B) = P(A)+P(B)
Axiomatic /modern Approach

 The event which includes all parts which are not in ‘A’ is called
a compliment of A and written as Ā
The shaded area on the diagram indicates Ā
Universe of Discourse – the set containing all elements under
discussion for a particular problem. In mathematics, this is called
the universal set and is denoted by U.
Venn diagrams can be used to represent sets and their
relationships to each other .
Complementation

SET THEORY
 Interpretation of statistical statements in terms of set theory:
 S = Sample Space = 𝐴 ∪ Ā
 A => A does not occur
 𝐴 ∩ Ā = 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒
 A U B = Event A occurs or B occurs or both A and B occurs.
 𝐴 ∩ 𝐵 = 𝐵𝑜𝑡ℎ 𝑒𝑣𝑒𝑛𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑐𝑐𝑢𝑟𝑠
 Ā ∩ 𝐵 = Neither A nor B occurs
 A ∩ 𝐵 = Event A occurs and B does not occur
 Ā ∩ 𝐵 = 𝐸𝑣𝑒𝑛𝑡 𝐴 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑐𝑐𝑢𝑟 𝑎𝑛𝑑 𝐵 𝑜𝑐𝑐𝑢𝑟𝑠.

1. The Addition theorem
•Mutually exclusive events
•Not mutually exclusive events
2. The Multiplication theorem
•Independent events
•Dependent events
Theories of Probability

The Addition Theorem
 It states that if two events A and B are mutually exclusive , the
probability of the occurrence of ‘A’ and ‘B’ is the sum of the
individual probability of A and B.
 P (A 0r B)= P(A)+P(B)
 Technically P(A ∪ 𝐵) = 𝑃 𝐴 + 𝑃 𝐵
Example : The probability that student X will s0lve a particular
problem is 1/8 and student Y will solve the same problem is ¼.
What is the probability that either A or B will solve the problem?
Solution: let’s A =event of solving the problem by X =P(A)=1/8
 B= event of solving the problem by Y =P(B)= ¼
 Therefore P (𝐴 ∪ 𝐵) =1/8 + ¼ = 4+8/32 =12/32=0.375

 Addition theorem for any two events (not mutually exclusive)
 If events are not mutually exclusive , then probability for A or B to
happen is the sum of their probabilities minus probability of both
to happen
 P (A 0r B)= P(A)+P(B) –P(A&B)
 P(A ∪ 𝐵) = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
 Example :find the probability of drawing an Ace or Club from a
normal deck of cards.
 P(A)= Event of Ace occuring =4/52 , P(B)= Event of club =13/52
 The event of club Ace occuring = P (𝐴 ∩ 𝐵) =1/52
 Then 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
 Ie. 4/52 + 13/52 – 1/52 = 16/52
The Addition Theorem

Multiplication Theorem
 It states that, if two events A and B are dependent , the
probability of the second event occuring will be effected by the
outcome of the first draw that has already occurred.
Symbolically P(A ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵/𝐴)
 Example : a bag contains 5 white balls and 8 black balls . One ball
is drawn at random from the bag and then it is not replaced.
Again another one is drawn. Find the probability that both the
balls drawn are white?
 Solution: P(A)= Event occuring white ball in first draw = 5/13
P(B/A )= Event occuring white ball in Second draw = 4/12
P of occuring white ball in both draws = P(A∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵/𝐴)
Ie , 5/13 x 4/12 =20/156

Multiplication Theorem
 It states that if two events A and B are independent , the
probability that they will both occur is equal to the product of
their individual probabilities.
 P (A&B)=P(A) X P(B)
 Technically P(A ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵)
P(A ∩ 𝐵 ∩ 𝐶) = 𝑃(𝐴). 𝑃(𝐵). 𝑃(𝐶)
 Example : a bag contains 5 white balls and 8 black balls . One ball
is drawn at random from the bag and then replaced. Again
another one is drawn. Find the probability that both the balls
drawn are white?
 Solution: P(A)= Event occuring white ball in first draw = 5/13
P(B )= Event occuring white ball in Second draw = 5/13
P of occuring white ball in both draws = P(A∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵)
Ie , 5/13 x 5/13 =25/169

 If two events A and B are dependent. Event B can occur only
when event A is known to have occurred or vice versa. The
probability attached to such an event is called conditional
probability and is denoted as P(B/A)
 If both events are dependent , Probability of B given A is
 P(B/A) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐴)
 If both events are dependent , Probability of A given B is
 P(A/B) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐵)
CONDITIONAL PROBABILITY

PRACTICE QUESTIONS
 Q2. When two dice are thrown, find the probability of getting
doublets ( same number on both dice)
 S= ( 6x6)=36

PRACTICE QUESTIONS
 Q1. Three coins are tossed simultaneously , find the probability
that 1) no head 2) one head 3) two heads
 4) At least two heads
 S= ( HHH, HHT, HTT, TTT,THT,THH,HTH,TTH)

Contribution of Sir Thomas Baye’s ,an English Philosopher
His contribution consists of a primarily a unique method for
calculating conditional probabilities.
Bayes theorem is based on the preposition that probabilities
should be revised when the information is available. Ie the
occurrence of one event to predict the occurrence of another.
The need for revising probabilities arise from a need to make
better use of available information.
This will help to reduce the element of risk involved in decion
making
 the probabilities before revision is called priori probabilities and
those after revision is called posterior probabilities
BAYES THEOREM

1. Under this principle , by assuming certain priori probabilities,
the posteriori probabilities are obtained. So the Baye’s
theorem are also called posteriori probabilities.
Rule : According to Baye’s theorem, the probability of event (A)
for a particular result of an investigation (B) may be found from
P(A/B) = P(A) P(B/A)
P(A) P(B/A)+P(Not A) P(B/Not A)
Similarly
P(B/A) = P(B) P(A/B)
P(B) P(A/B)+P(Not B) P(A/Not B)
BAYES THEOREM

EX: A is known to tell the truth five times in cases out of six. He the
states that a white ball was drawn from a bag 0f 9 black and one
white ball. What is the probability that the white ball is really
drawn?
Solution :
The probability that the white ball is drawn =1/10
The Probability that white ball is drawn and A told the truth
= 1/10 x 5/6
The probability that a black ball was drawn and A told a lie
=9/10 x 1/6
The probability that a white ball was drawn P(A) ,given B
P(A/B) = P(A) P(B/A) = 1/10 X 5/6 = 5
P(A) P(B/A)+P(Not A) P(B/Not A) (1/10 X5/6) + ( 9/10X1/6) 14
BAYES THEOREM

 CLT in probability theory states that , given certain conditions,
the arithmetic mean of a sufficiently large number of iterates
of independent random variables, each with a well defined
expected value and well defined variance, will be
approximately normally distributed, regardless of the
underlying distribution.
 CLT states that the distribution of sample approximates a
normal distribution (also known as a bell curve) as the sample
size becomes larger, assuming all samples are identical in size
regardless of the population distribution shape
 Examples : laboratory measurement errors are usually modeled
by normal random variables.
 in finance, percentage of changes in prices of some assets are
sometimes modeled by normal random variables
Central limit theorem

 Fundamental principle
 If one thing can be done in ‘m’ different ways and a second
thing can be done in ‘n’ different ways. Thus the two things can
be done ‘m’ ‘n’ different ways
 Factorial symbol
 nỊ = n factorial denotes the product
 nỊ = n(n-1)Ị
 (n+1)Ị = (n+1)nỊ
 Ex: 6Ị = 6x5x4x3x2x1= 720
Permutation and combination

 It refers to different arrangements of objects in a set when all
elements are different and distinguishable. The arrangements
are done without repetition of an individual object
 It refers to separate arrangements of different objects
contained in a set of elements.
Example : if a factory owner receives 3 different machines, A B
and C. how many ways it can be ?
ABC, ACB, BCA,BAC,CBA,CAB
nỊ= 3Ị= 3x2x1=6 times
Permutation

 When we have ‘n’ different objects and ‘r’ space is to be filled, then
by the multiplication rule, the space can be filled in as follows
 n(n-1)(n-2)(n-3)…….
 Let npr = n(n-1)(n-2)…..(n-r+1)
 By using factorial symbols
 Npr = n(n-1)(n-2)…(n-r+1)(n-r)Ị = nỊ
(n-r)Ị (n-r)Ị
Ex: indicate how many ways 5 letter words can be formed from the
letters in the word ‘EQUATIONS’
SOLUTION; here n= 9, r=5
 npr = nỊ = 9p5= 9Ị = 9x8x7x6x5x4x3x2x1 = 9X8X7X6X5
(n-r)Ị (9-5)Ị 4x3X2x1
= 15,120
Permutation

 A combination is a selection if objects.
 Combination is a grouping or selection of all or part of a number of
things without reference to the arrangement of the thing selected.
The number of combination of objects are different from their
permutation
 The combination of the letter A B and C are ABC
 The total number of possible combinations of a set of objects taken
at a time is 1. only one combination is possible for a set.
 The number of combinations of different objects taken ‘r’ at a time
is denoted as ncr = nỊ
rỊ (n-r)Ị
In combination ,we are not concerned with the arrangements of
objects . In permutations , a combination can give rise to several
permutations
Ex: combination ABC , Permutation: ABC,ACB,BAC,BCA,CAB,CBA
combination

 How many different sets of 5 students can be chosen out of 20
qualified students to represent a college in a quiz competion?
 n = 20 (no. of qualified students )
 r = 5 ( no. of students to be selected at a time)
 Ncr = nỊ
rỊ (n-r)Ị
 20c5 = 20Ị = 15504
5Ị (20-5)Ị
combination

 In how many ways can 4 red balls be drawn from a bag
containing 10 red balls
 n = 10 (no. of red balls )
 r = 4 ( no. of red balls to be selected at a time)
 Ncr = nỊ
rỊ (n-r)Ị
 10c4 = 10Ị =210
4Ị (10-4)Ị
combination