PROF. AHMAD NAWAZ NASIR
For Graduate
And
1
Post
PROBABILITY
Graduate
Students
PROF. AHMAD NAWAZ NASIR
Senior Subject Specialist
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PROF. AHMAD NAWAZ NASIR
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PROBABILITY
PROBABILITY
SET:-
A
set is any well-defined collection or list of objects. The objects that make up a
set (also known as the elements or members of a set) can be anything: numbers,
people, letters of the alphabet, other sets, and so on
SUBSET:-
I
f every member of set A is also a member of set B, then A is said to be a subset
of B, written A ⊆ B (also pronounced A is contained in B).
If A is a subset of, but not equal to, B, then A is called a proper subset of B, written
A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).
A is a subset of B
Example:
The set of all men is a proper subset of the set of all people.
{1, 3} ⊆ {1, 2, 3, 4}.
{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set and every set is a subset of itself:
Power SET:
T
he power set of a set S is the set of all subsets of S. It is denoted by P(S).If S=
{1, 2, 3}. Then Power set is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}.
Null set:- A set that has no element is called an empty or null set. It is denoted by
 or { }.
Unit Set:-
If a set contain only one element, it is called a unit or singleton set.
Universal set:-
T
he set containing all objects or elements and of which all other sets are subsets.
OR In set theory, a universal set is a set which contains all objects, including
itself.
Venn diagram:-
A
Venn diagram or set diagram is a diagram that shows all possible logical
relations between a finite collection of different sets
Union of Set:-
T
he union of Two sets A and B is the set
of all elements which belongs to A or to B or both.
It is denoted by A ∪ B.
A B =  X / x A or x  B
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PROBABILITY
Intersection of set:-
T
he intersection of two sets A and B is the set of all
elements that belongs to both A and B. It is denoted by A ∩ B.
If A ∩ B = ∅, then A and B are said to be disjoint.
B = X/XAand XB
A
Difference of sets:-
T
he difference of two sets A and B denoted by A – B(A \ B) , is the set of all
elements belongs to “A” but do not belongs to “B”.
T
he complement of set “A” is the set of elements which do not belongs to A.
It is denoted by A or A or Ac .
Complement Set:-
Sure Events:-
T
he sample space is subset of itself, hence it is also an event. It always occurs and
is known as the certain or sure event. A sure event is the one that contains the
whole sample space. For example, in our experiment of throwing a dice
Impossible events:-
A
n impossible event is the opposite case, when the event does not contain any
element of the sample space.
For example, the event
Ordered Pairs:-
A="get a 7" on throwing a dice of six faces.
O
rdered pair (a, b) is a pair of mathematical objects. The order in which the
objects appear in the pair is significant: the ordered pair (a, b) is different from
the ordered pair (b, a) unless a = b. Two numbers written in a certain order. Usually
written in parentheses like this: (4, 5) Can be used to show the position on a graph,
where the "x" (horizontal) value is first, and the
"y" (vertical) value is second.
Here the point (12, 5) is 12 units
along, and 5 units up.
Product Sets:-
T
he sets A and B, the Cartesian
product A × B is the set of all
ordered pairs (a, b) where a ∈ A and b ∈ B. Products can
be specified using set-builder notation, e.g.
A table can be created by taking the Cartesian product of a set of rows and a set of
columns. If the Cartesian product rows × columns are taken, the cells of the table
contain ordered pairs of the form (row value, column value).
Experiment:-
The process of obtaining the observations is called experiment.
Random Experiment:-
A
n experiment which produces different results even though it is performed
under the same condition is called random experiment. For Example: - Tossing of
a coin, throwing of a die. It has following properties.
(i) The experiment may be repeated for any number of times.
(ii) The experiment always has two or more possible outcomes.
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PROBABILITY
(iii) The outcomes are unpredictable. (‫ جو پہلے سے نظر نہ آئے‬٬‫)ناقابل پیشن گوئی‬
Sample Space:-
T
he set or collection of all possible outcomes of the random experiment is called
Sample space. It is denoted by S. Throwing of a die.
S = {1, 2, 3, 4, 5, 6}, Tossing a pair of coins. S = { HH, HT, TH, TT}
Sample Points:The elements of sample space are called sample points.
Event:-
A
n event is any collection of outcomes of an experiment.
Formally, any subset of the sample space is an event. Any event which consists of
a single outcome in the sample space is called an elementary or simple event. Events
which consist of more than one outcome are called compound events.
Equally Likely:-
T
wo events A and B are said to be equally likely if they have same chance of
appearance. For Example, if we toss a fair coin, the head is as likely to occur as
the tail.
Mutually Exclusive:-
T
wo events A and B are said to be mutually exclusive if they cannot appear
together. For example, we toss a coin, if head occurs, the tail cannot occur.
T
wo events A and B are said to be non-mutually exclusive if they can appear
together. For Example, King card and diamonds cards are not mutually exclusive.
Non Mutually Exclusive:Exhaustive Event:-
T
hose outcome which form the entire sample space are known as exhaustive )
events.
For Example, S = {1, 2, 3, 4, 5, 6} , A = {1, 3, 5}, and B = {2, 4, 6}, then A U B = S
Probability:- (statistics)
I
t is a measure or estimate of the degree of confidence one may have in the
occurrence of an event, measured on a scale from zero (impossibility) to one
(certainty). It may be defined as the proportion of favorable outcomes to the total
number of possibilities
Probability may be defined into two different approaches, objective and subjective
approach.
Objective approach:-
T
he probability that an event will occur based an analysis in which each measure is
based on a recorded observation, rather than a subjective estimate. Objective
probabilities are a more accurate way to determine probabilities than observations
based on subjective measures, such as personal estimates.
Classical definition:-
If a random experiment produces “n” mutually exclusive and equally likely outcomes
in S and “m” are the favorable outcomes of an event A, then the probability of A
Favorable outcomes in A
m
=
defined as: P(A) =
Total possible outcomes in S
n
It has following shortcomings.
(i) In this definition, we use the term equally likely which means equally probable.
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It means that we have advance or priori knowledge of probability.
(ii) This definition is not applicable if assumption of equally likely does not hold.
(iii) This definition becomes fail when number of possible outcome is infinite.
Relative Frequency or Posteriori Definition:-
I
f an experiment is repeated n times, and event A occurs m times, then the
relative frequency of the event A is defined to be
m
P(A) =
n
If an experiment is repeated many, many times without changing the experimental
conditions, the relative frequency of any particular event will
m
P(A) = Limit n  
n
This definition assumes that as n   , the ratio m/n becomes stable at the numerical
value of P(A). It has following shortcomings.
(i) The experiment may change from trial to trial.
(ii) The experiment is repeated for a finite number of times, therefore the ratio m/n
may not be unique.
It is more useful for practical problems, it’s an advantage.
Axiomatic Approach:-
L
et a sample space “s” has the sample points E1 ,E 2 , E 3 ,............, E n
Then we assign a real number of each sample point, P(Ei) is called probability of
E. But it must satisfy the following basic axioms.
(i) 0  P(Ei )  1
(ii) P(S) = 1
(iii) If A and B are mutually exclusive events, then P( A B)  P( A)  P( B)
Subjective Probability:-
A
subjective probability describes an individual's personal judgment about how
likely a particular event is to occur. It is not based on any precise computation
but is often a reasonable assessment by a knowledgeable person.
For Example:- There is 50%chance won of Pakistani team, there is 30% chance will
rain tomorrow. It has a disadvantage that two or more persons faced with the same
evidence may arrive at different probabilities. It has a advantage may be applied to
those real world situations where neither an equally likely nor relative frequency
approach is possible.
Probability of an event:-
If an event “A” is defined in sample space “S” then probability of “A” is defined as
number of sample points in “A” is divided by total
Number of sample points in A
m
number of sample points in “S”, then P(A) =
=
Number of sample points in S
n
Addition Law of Probability:Statement:- If two events A and B are not mutually exclusive, then the probability
that at least one of them occur, is given by the sum of the separate probabilities of
events A and B minus the probability of the joint event A∩B.
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P(AUB) = P(A) + P(B) - P(A B)
Proof:- According to venn diagram,
AUB  A U (A B)
Taking probability on both sides
P( AUB )  P(A U (A B))
A and (A B) are mutually exclusive events
P( AUB )  P(A)  P (A B)  (1)
B = (A B) (A B)
Taking probability on both sides
P(B) =P[ (A B) (A B)]
(A B) and (A B) are mutally exclusive events
P(B) = P (A B)  P (A B)
P (A B)  P(B) - P (A B)
putting in equation 1 and we get
P( AUB )  P(A)  P(B) - P ( A B)
Addition Law For Mutually Exclusive Events:Statement:- If two events A and B are mutually exclusive, then the probability that
at least one of them occur, is given by the sum of the separate probabilities of
events A and B, then
P(AUB) = P(A) + P(B)
Proof:- Let n be the number of sample points in a sample space S. Let A and B be the
two mutually exclusive events in the sample space S, such that events A and B
contains m1 and m2 sample points respectively. Then AUB will contain the sample
points belongs to either A or B. Then
A B 
The sample points AUB will be m1 + m2
m  m2 m1 m2
P(AUB) = 1


 P( A)  P( B)
n
n
n
P( AUB)  P( A)  P( B)
Corollary:If A1 ,A 2 ,A 3 ,.........A k are mutually exculisive events, then the probability that one of them occurs,
is the sum of the probabilities of the separate events, then
P  A1
A2
A3 ....... Ak   P( A1 )  P( A2 )  ...............  P( Ak )
If A and B are any two events, then
P( AUB)  P( A)  P( B)
When P(A
B)  0
This result is known as Boole ' s inequality
Conditional Probability:-
T
he sample space for an experiment is change when some additional information
about the outcomes of an experiment is received. The new sample space is called
reduced sample space and probabilities associated with such a reduced sample
space are called conditional probability.
If A and B are two events in a sample space S and if probability of B is not equal to
zero, then the conditional probability of the event A given that the event B has
occurred, then
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 B = P(AP(B)B)
P A
 A =
P B
,
P(B) > 0
P(A B)
,
P(A)
P(A) > 0
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PROBABILITY
Multiplication Law for two dependent Events:Statement:- If A and B are two not independent events, then the probability that
both A and B occur is equal to the probability that one of the events will occur
multiplied by conditional probability that the other event will occur given that the
first event has occurred.
P(A B)
P A
=
, P(B) > 0
B
P(B)
P(A B)
P B
=
, P(A) > 0
A
P(A)
 
 
Proof:Let A and B are two dependent events defined in sample space S. we know by
definition of conditional probability of A given B is:
Multiplying by P(B) on both sides
 B  P(B) = P(AP(B)B)  P(B)
P  A   P( B) = P(A B)
B
 P(A B)  P  A B  P( B)
P(A B)  P( B)  P  A 
B
P A
;
P(B) = 0
;
P(B) = 0
;
P(B) = 0
;
P(B) = 0
Similarly,
 A =
P(A B)
P(A)
Multiplying by P(A) on both sides
P B
 A   P( A) = P(AP(A)B)  P( A)
P  B   P( A) = P(A B)
A
 P(A B)  P  B A   P( A)
P(A B)  P( A)  P  B 
A
P B
;
P(A) = 0
;
P(A) = 0
;
P(A) = 0
;
P(A) = 0
;
P(A) = 0
Multiplication Law for two Independent Events:Statement:- If A and B are two independent events, then the probability that both
A and B occur is equal to the probability that one of the events will occur multiplied
by other event will occurs.
Proof:- Let A and B are two independent events defined in independent sample space
consisting of n1 and n2 sample points. Let m1 and m2 sample points are favorable to the
occurrence of A and B.
m
m
P(A) = 1 AND P( B)  2
n1
n2
A and B are independent events therefore possible and favorable sample points to the
occurrence of (A∩B) will be n1n2 and m1m2 respectively. Thus
mm
m m
P( A B)  1 2  1 2  P( A).P( B)
n1n2
n1 n2
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PROBABILITY
Q:- Elaborate the statement that “two mutually exclusive events need not to be
equally likely” by giving example.
Solution:-
Two mutually exclusive events need not be equally likely because there are certain
situations in which events remain mutually exclusive but the property of equally
likely vanishes. For Example if an unbalanced coin is tossed the occurrence of head
and tail is not equally likely but head and tail is mutually exclusive events.
Q;- Explain independent and dependent events.
Solution:-
Independent Events:-
Two events are said to be independent if occurrence of one event does not affect
the occurrence of the other. For Example in the experiment of tossing a coin and a
die, the occurrence of tail on the coin is not affected by the occurrence of 6 on the
die.
If A and B are two independent event if P(A∩B) = P(A)P(B)
Dependent Events:- Two events A and B are called dependent if the occurrence of
one event affected the occurrence of the other. For Example, let a bag contains 4
green and 5 red balls and if 2 balls are drawn one by one without replacement. Then
the probability of green or red ball is affected by either it is selected on the 1st or
2nd draw.
Q:- Discuss the idea of independent and mutually exclusive events.
Solution:-
Two mutually exclusive events A and B are independent if and only if P(A)P(B) = 0,
which is true when P(A) = 0 or P(B) = 0. If both events A and B have non zero
probability, they must have a common sample point. Thus two events that are
independent can never be mutually exclusive. Two events that are mutually exclusive
are also dependent events, and two events that are non-mutually exclusive, may either
be independent or dependent events.
Independent Repeated Trials with two outcomes:-
If the probability of an event A occurring in a single trial is p, then the probability
of its occurring k times in n independent trials is given by
n
P ( A)   k  p k q n  k


q=1–p
NOTE:-
1 – Probability never becomes negative.
2 – The range of probability is from 0 to 1.
3 – Total probability is always equal to 1.
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All Possible out comes with pair of dice = 62
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  
S 

 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 
         
  
All possible outcomes with two coins = 4
S  HH , HT , TH , TT 
All possible outcomes with tree(3) coins = 8
S HHH , HHT , HTH , THH , HTT , THT , TTH , TTT 
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PROBABILITY
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