explains probability through concept of modern set theory.

2. PROBABILITY

3. • Probability:
the word probability has two basic meanings:
a) a quantitative measure of uncertanity.
b) a measure of degree of belief in a particular
statement or problem.
probability is basically the measure of likeliness that
an event will occur.

4. • It is quantified as a number between 0 and 1
• 0 indicates “impossibality” & 1 indicates “certanity”
• The higher the probability of event, the more certain
we are that event will occur.
• Probability has wider field of applications and is used
to make intelligent decisions in Economics,
Management, sociology , Engineering etc. where risk
and uncertanity are involved.
• Probability can be best understood by application of
“Modern Set Theory”

5. Modern set theory
• SET:
A set is any “well defined collection” or list of
“distinct objects” .
the objects that are in a set are called element or
member of a set.
Example:
A={1,2,3,4,5}

6. • A set which has no element in called an empty set. It
is denoted by φ or { }
• A set may be specified in two ways:
1) roster method.
A={1,3,5,7,9,11}
2) set builder method.
A={xІx is an odd no and x<12}

7. • Subset:
a set that consist of some elements or member of
another set is called subset of that
set.
if B is a subset of A then it is denoted as B⊂ A
A set with n elements will produce 2ʱ subsets ,
including universal set and empty set.

8. • Venn diagram:
A diagram representing
mathematical or logical sets
pictorially as circles or closed
curves within an enclosing
rectangle (the universal set),
common elements of the sets
being represented by
intersections of the circles.
Venn diagram are used to
represent sets and subsets in
pictorial way and to verify
the relationship among set
and subsets.

9. • OPERATIONS ON SETS:
there are 4 basic operations
a) union
b) intersection
c) difference
d) complement

10. • Union:
Elements in at least one of the two sets:
denoted by AUB
AB = { x | x  A  x  B }
A BAB

11. • INTERSECTION:
Elements in exactly one of the two sets:
denoted by A∩B
AB = { x | x  A  x  B }
A BAB

12. • DIFFERENCE:
Elements in first set but not second:
denoted by A-B
A-B = { x | x  A  x  B }
A
BA-B

13. • COMPLIMENT:
A = { x | x  A }
Elements not in the set (unary operator):
A
U
A

14. Set Properties
• Commutative Laws:
• Associative Laws:
• Distributive Laws:
14
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)(
)(
)()()(
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CBACBAb
CBACBAa

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)()()()(
)()()()(
CABACBAb
CABACBAa



15. Set Properties
• Double Complement Law:
• De Morgan’s Laws:
• Absorption Laws:
15
AA cc
)(
ccc
ccc
BABAb
BABAa


)()(
)()(
ABAAb
ABAAa

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)()(
)()(

16. Partitions
• Definition: A collection of nonempty sets
{A1, A2, …, An} is a partition of a set A if
1. A = A1  A2  …  An
2. A1, A2, …, An are mutually disjoint.
• Examples:
1) Let S0={n  Z | n=3k for some integer k}
S1={n  Z | n=3k+1 for some integer k}
S2={n  Z | n=3k+2 for some integer k}
Then {S0, S1, S2} is a partition of Z.
16

17.  Cartesian Product:
Given two sets A and B, the set of all ordered
pairs of the form (a , b) where a is any element
of A and b any element of B, is called the
Cartesian product of A and B.
Denoted as A x B
A x B = {(a,b) | a A and b B}

18. (1) TREE DIAGRAMS
A tree diagram is a diagram used to show the
total number of possible outcomes in a
probability experiment.

19. (2) THE FUNDAMENTAL
COUNTING PRINCIPLE
The Fundamental Counting Principle uses
multiplication of the number of ways each
event in an experiment can occur to find
the number of possible outcomes in a
sample space.

20. How many outfits are possible from a pair of jean or shorts and a
choice of yellow, white, or blue shirt?
1)
2)
jeans SHORTS
YELLOW WHITE BLUE YELLOW BLUE WHITE
No of jeans & shorts No of Shirts Total Outcomes
2 3 06

21. RANDOM EXPERIMENT
• The term “experiment” means a planned activity and
a process whose result yield a set of data.
• An experiment which produces different results even
though it is repeated a large no of times under
essentially similar conditions is called an “Random
Experiment”.
• A single performance of an experiment is called
“Trial”
• The result obtained from a trial is called an
“outcome”.

22. PROPERTIES OF RANDOM EXPERIMENT
A random experiment has three properties:
a) The experiment can be repeated , practically or
theoretically, any no of times.
b) The experiment always has two or more possible
outcomes.
c) The outcome of each repetition is unpredictable.

23. SAMPLE SPACE
• A set consisting of all possible outcomes that can
result from a random experiment is called “sample
space”
• It is denoted by S.
• Each possible outcome is a member of sample space
and is called “Sample Point”.
• For example, if the experiment is to throw a die and
record the outcome, the sample space is
S={1,2,3,4,5,6}

24. • A sample space that contains a finite no of
sample points is called “Finite sample space”
• If sample points can be placed in one-to-one
correspondence with the positive integer or if
it is finite then it is called “Discrete Sample
Space”.
• If it satisfies neither of these criteria it is called
“continuous sample space”

25. EVENTS
• An event is an individual outcome or any number of
outcomes of a random experiment.
• Any subset of sample space of the experiment is
called an event.
• An event that contains exactly one sample point is
called “simple event”.
• Even that contains more than one sample point is
called compound event.

26. • An event A is said to occur if and only if the outcome
of the experiment corresponds to some element of A.
• The event “not-A” is denoted by A and called
negation or complementary of A.
• For example, the compliment of “Head” is “Tail” for
tossing one coin.
• A sample space consisting of n sample points can
produce 2ⁿ.

27. • EXAMPLE:
consider, S={a , b , c}
no of subset=2ⁿ=8
Possible subsets are:
φ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}
 Each of these subsets is an event.
 Event {a,b,c} is sample space itself & called sure
event.
 Empty set φ is also an event and is called impossible
event.

28. MUTUALLY EXCLUSIVE EVENTS
• Two events A and B of a single experiment are said to
be mutually exclusive or disjoint if and only if they
cannot occur at the same time.
• Three or more events originating from the same
experiment are mutually exclusive if pair-wise the are
mutually exclusive.

29. EXHAUSTIVE EVENTS
• Events are said to be exhaustive when union
of mutually exclusive events is the sample
space itself.
• A group of mutually exclusive and exhaustive
events is called partition of sample space.
• For example ,events A and A

30. EQUALLY LIKELY EVENT
• Two events are said to be equally likely when one
event is as likely to occur as the other. In other words
, each event is as likely to occur as the other.
• For example, when a fair coin is tossed , probability
of both head and tail is ½.

31. COUNTING SAMPLE POINTS
• When no of sample points in a sample space are very
large then it becomes inconvenient to list all of them.
So we use following rules for counting sample points:
1) rule of multiplication.
2) rule of permutation.
3) rule of combination.

32. Rule of multiplication
• Consider a compound experiment consisting
of two experiments . If first experiment has
‘m’ distinct outcomes and corresponding to
each outcome of first ,there are ‘n’ distinct
outcome of second experiment , then :
total outcomes= m x n

33. RULE OF PERMUTATION
• Permutation is any ordered subject from a set
of n distinct object.
• The no of permutations of ‘r’ objects ,
selected in a definite order from a set of ‘n’
distinct objects is denoted by nPr
nPr = n! / (n - r)!

34. RULE OF COMBINATION
• A combination is any subset of r objects ,
selected without regard to their order , from a
set of ‘n’ distinct objects.
• Combination is denoted by nCr
nCr = n! / r!(n - r)!

35. DEFINITIONS OF PROBABILITY
PROBABILITY
OBJECTIVE
SUBJECTIVE
CLASSICAL
DEFINITION
THE RELATIVE
FREQUENCY
AXIOMATIC
DEFINITION

36. CLASSICAL DEFINITION OF
PROBABILITY
• If a random experiment can produce ‘n’ mutually
exclusive and equally likely outcomes and if ‘m’ out
of these outcomes are favorable to occurrence of
certain event ‘A’ , then probability of event A , is
defined as the ratio m/n.
P(A)=(no of favourableoutcome)/total outcome
 EXAMPLE:
a) The roll of a die: There are 6 equally likely outcomes. The probability of
each is 1/6.
b) The toss of two coins: The four possible outcomes are {H,H}, {H,T},
{T,H} and {T,T}. The probability of each is 1/4.

37. LIMITATION OF CLASSICAL DEFINITION
• This definition is only applicable if events are equally
likely.
• No. of possible outcomes should always be finite.

38. THE RELATIVE FREQUENCY DEFINITION
• When a random experiment is repeated a large no of
times , say n , under identical conditions and if an
event A is observed to occur ‘m’ times ,then
probability of event A is defined as the limit of R.F
m/n as n tends to infinity.
P(A)=lim n→∞ (m/n)
when n increases infinitely, the ratio m/n tends to
become stable.

39. AXIOMATIC DEFINITION
• In mathematics, an axiom is a result that is accepted without the need for
proof.
• In this case, we say that this is the axiomatic definition of probability
because we define probability as a function that satisfies the three axioms
given below.
If we do a certain experiment, which has a sample space Ω, we define the
probability as a function that associates a certain probability, P(A) with
every event A, satisfying the following properties.
 The probability of any event A is positive or zero. Namely P(A)≥0.
The probability measures, in a certain way, the difficulty of
event A happening: the smaller the probability, the more difficult it is to
happen.

40.  The probability of the sure event is 1. Namely P(Ω)=1. And so, the
probability is always greater than 0 and smaller than 1: probability zero
means that there is no possibility for it to happen (it is an impossible
event), and probability 1 means that it will always happen (it is a sure
event).
 The probability of the union of any set of two by two incompatible
events is the sum of the probabilities of the events. That is, if we have,
for example, events A,B,C, and these are two by two incompatible, then
P(A∪B∪C)=P(A)+P(B)+P(C).

41. SUBJECTIVE PROBABILITY
• Also known as personalistic probability.
• It is the measure of strength of a person’s belief
regarding the occurrence of an event A.
• This is based on whatever evidences are available to
the individual.
• The disadvantage of subjective probability is that two
or persons faced with the same evidence ,may arrive
at different probabilities.

42. LAWS OF PROBABILITY
1. 0 ≤ P(E) ≤1 The probability of an event E is between
0 and 1 inclusive.
2. P( ) = 0
The probability of an empty set is zero.
Consequence: IF P(A ∩ B) = 0 then
it implies A and B are mutually exclusive.
3. P(S)=1 The probability of the sample space is 1.

43. ADDITION LAW
• The probability that Event A or Event B occurs is equal to
the probability that Event A occurs plus the probability
that Event B occurs minus the probability that both
Events A and B occur.
for mutually exclusive events:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(AUB)=P(A)+P(B)

44. LAW OF COMPLEMENTATION
• The probability that event A will occur is equal to 1
minus the probability that event A will not occur.
P(A) = 1 - P(A')

45. PROBABILITY OF SUBEVENT
If A and B are two events such that A ⊂ B then
P(A) ≤ P(B)

46. THEOREM
• If A, B, C are any three events in a sample
space S, then the probability of at least one of
them occurring is given by :
P(AUBUC) = P(A) + P(B) + P(C) – P(A∩B) –P(B∩C) – P(A∩C) +P(A∩B∩C)

47. THEOREM
• If A and B are any two events defined in a
sample space S, then :
P(A ∩ B’) = P(A) – P(A ∩ B)