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The above circles have same permutations.

Example:
Arrange 5 different trees in a circle:

Solution:
n = 5 trees
   = (n – 1)!
   = (5 – 1)! = 4! = 24 ways

Combinations: A combination is a group of items without regard to the
arrangement of items. ABC and BCA are two different permutations but are
same combinations of three letters.

The number of combination of n distinct items taken r at a time, denoted
by   EMBED Equation.3     , is:

  EMBED Equation.3
Example:
In how many ways five students can be selected from a group of 12
students?

Solution:
n = 12; r = 5

  EMBED Equation.3

Basic Concepts of Probability Theory:

Possibility Space,
Event,
Complementary Event,
Mutually Exclusive Events,
Composite Events, and
Joint Events.

Possibility Space: Also known as ‘sample space’ or ‘outcome space’. A
possibility space is a set of all possible outcomes of an experiment and
is denoted by S.

If an experiment consists of a toss of a fair die and the numbers are of
interest, the possibility space would be:

S = {1, 2, 3, 4, 5, 6}

If the interest is whether the number is even or odd, the possibility
space would be:

S = {even, odd}

A possibility space may be represented by a rectangle.   The number of
outcomes in the possibility space is denoted by n(s).

Event: A subset of a possibility space is called an event and is usually
denoted by first few capital letters A, B, C, ……… For example, a coin is
tossed, the sample space is S = {H, T} and the subset A = {H} is the
event when a head occurs.

Take another example, two coins are tossed. The sample space is S = {HH,
HT, TH, TT} and the subset B = {HH, HT, TH} is the event that at least
one head appears when two coins are tossed.

An event may be represented by a circle inside the rectangle of the
possibility space.

An event is further divided into the following:

Simple Event: is the subset if it contains only one outcome of the
possibility space.
Compound Event: is the subset if it contains more than one outcomes of
the possibility space.
Null Event: is a subset containing no outcomes. It is also called
‘impossible event’.
Sure Event: It is also called ‘certain event’.    It is a subset containing
all outcomes of the possibility space.

Number of possible events = 2n

Example:
Two coins are tossed. List all the possible events or subsets of the
possibility space.

Solution:
S = {HH, HT, TH, TT}
n(s) = 4

Number of possible events = 2n = 24 = 16 events

Possible Events of two tossed coins:




  E1 = { }             Null Event
E2 = {HH}
 E3 = {HT} Simple Events
E4 = {TH}
E5 = {TT}
 E6 = {HH, HT}
E7 = {HH, TH}
E8 = {HH, TT}
E9 = {HT, TH}
E10 = {HT, TT}
 E11 = {TH, TT} Compound Events
E12 = {HH, HT, TH}
E13 = {HH, HT, TT}
E14 = {HH, TH, TT}
E15 = {HT, TH, TT}
 E16 = {HH, HT, TH, TT}      Sure Event

Complementary Event: For an event A, the complementary event is defined
as a set of those outcomes of the possibility space which are not in A.
The complementary event of A is written as   EMBED Equation.3      (not
A).

For example, the complementary events of E5 = {TT} are    EMBED Equation.3
     .   It is diagrammatically shown as below:




Mutually Exclusive Events: Two events A and B are mutually exclusive if
they have no outcomes in common and therefore they cannot happen
together. Mutually exclusive events are also known as disjoint events.

Non-mutually exclusive events are the vice versa of the above definition
and are also known as over-lapping events.




For example, in case of two tossed coins, E6 and E7 are not mutually
exclusive events, because both events have an outcome HH in common.
However, E6 and E11 are two mutually exclusive events because no single
outcome is common.

Composite Events: For two events A and B, a composite event is defined as
a set of outcome of either A or B or both A and B and therefore at least
one of two events must occur. The composite event of A and B is written
as AUB, or A or B.

For example, the composite event of E6 and                E 7    i s      a s
 f o l l o w s :

 E 6     U    E 7       =   { H H ,    H T ,     T H }

 J   o i n t   E    v   e n t s :     F o r     t w o     e   v e n   t s ,     A   a n   d
 B   ,   a   j o    i   n t   e v e   n t     i s   d e   f   i n e   d   a s     a   s   e t
 o   f   c o m m    o   n   o u t c   o m e   s   o f     A     a n   d   B     a n d
 t   h e r e f o    r   e   b o t h     t h   e   e v e   n   t s     m u s t     o c c   u r
 t   o g e t h e    r   .     T h e     j o   i n t   e   v   e n t     o f     A   a n   d
 B     i s   w r    i   t t e n   a   s       A   a n d       B       o r       A )"B     .

 F o r       e x a m p l e ,     t h e   j o i n t            e v e n t     o f     E 6
 a n d       E 7   i s   a s     f o l l o w s :
 E 6      )"       E 7       =       { H H }

 T a k e   a n o t h e r   e x a m p l e ,   t h e                             j o i n t
 e v e n t   o f   E 2   a n d   E 1 5   i s   a s                             f o l l o w s :

 E 2   )" E 1 5                  =    {          }            ( i . e . ,          N u l l
 E v e n t )

 P r o b a b i l i t y                    T h e o r y :

 A       p r   o   b   a b   i   l   i t y       i   s   a   n u m e r i   c   a   l   m e a s u r e
 o   f     t   h   e     l   i   k   e l i   h   o   o d   ( o r   c h a   n   c   e )   t h a t   a
 p   a   r t   i   c   u l   a   r     e v   e   n   t   w i l l   o c c   u   r   .
 T   h   e     p   r   o b   a   b   i l i   t   y     o f   a n y   e v   e   n   t   m u s t
 s   a   t i   s   f   y     t   h   e   f   o   l   l o w i n g   t w o
 c   o   n d   i   t   i o   n   s   .

 N o   p r o b a b i l i t y                         i s   n e g a t i v e ,         P ( E v e n t )
 e" 0 .
 N o   p r o b a b i l i t y                         i s   g r e a t e r       t h a n       o n e ,
 P ( E v e n t )   d" 1 .

 T h e r e   a r e   t h r e e   d i f f e r e n t                             a p p r o a c h e s
 to assign probabilities to the events:

Classical or Mathematical Approach
Empirical or Relative Frequency Approach
Subjective Approach

Classical Approach: It is the approach in which probabilities are
assigned to the events before the experiment is actually performed and
therefore, such probabilities are also called ‘a priori’ probabilities.

If the possibility space of the experiment is finite, and if each outcome
of the possibility space is equally likely to occur, then the probability
of event A:

     EMBED Equation.3

It is also referred to as ‘axiomatic function of probability’.

Example:
Two coins are tossed once, what is the probability that two heads will
appear?

Solution:
S = {HH, HT, TH, TT}
n(S) = 4
Event = Two head appear
i.e., n(A) = 1

Now the probability of event A is as calculated below:
  EMBED Equation.3

Relative Frequency Approach: This approach is applied when the
possibility spaces are infinite, or the outcomes cannot be assumed
equally likely.

If an experiment is represented ‘n’ times under uniform conditions and if
‘m’ times the outcome of the experiment is in favour of an event A, then
the ratio   EMBED Equation.3      approaches the probability of the event
A as ‘n’ approaches infinity.

  EMBED Equation.3

The ratio   EMBED Equation.3      is considered as an estimate of the
actual probability of event A and normally this estimate is called the
probability of the event A and is written as:

  EMBED Equation.3

Since in this approach probabilities are assigned after performing a
large experiment, therefore, they are also known as ‘a posteriori’
probabilities.

Example:
A die has been rolled 360 times and ‘Six’ has been observed 63 times.
Estimate the probability of occurring a ‘Six’ when the die is to be
rolled once again.

Solution:
m = 63
n = 360

  EMBED Equation.3

Subjective Approach of Probability: Subjective probability can be defined
as the probability assigned to an event by an individual, based on
whatsoever evidence is available. This evidence may be in the form of
relative frequency of past occurrences, or it may be just an educated
guess.

Probability of Complementary Events:
If A and   EMBED Equation.3      are complementary events in a
probability S, then:

  EMBED Equation.3

Example:
A coin is tossed 3 times.   Find the probability of getting at least one
tail.

Solution:
Number of possible outcomes in S = n(S) = 23 = 8
Let A = At least one tail appears
Then   EMBED Equation.3      = No tail appears = All heads appear
     EMBED Equation.3   (the only outcome with all heads)
     EMBED Equation.3

Independent and Dependent Events:

When two events are given, the occurrence of the first event may or may
not have an effect on the occurrence of the second event.
When the occurrence of one of the two events has no effect on the
probability of the occurrence of the other event, the two events are
called ‘Independent Events’.
On the other hand, when the occurrence of first event has some effect on
the probability of occurrence of second event, the second event is said
to be ‘Dependant’ on the first event.

Conditional Probability:
If there are two events A and B such that the probability of event B
depends on the occurrence or non-occurrence of the event A, then the
probability of event B occurs when the event A occurs is called the
‘conditions probability of event B given event A’ and is written as:

     EMBED Equation.3

and

     EMBED Equation.3

If two events are independent then:

P(B/A) = P(B)           (independent events)

Example:
A black card   i s   d r a w n   f r o m   a n   o r d i n a r y
 d e c k   o f   5 2   p l a y i n g   c a r d s .     W h a t   i s
 t h e   p r o b a b i l i t y   t h a t   i t   i s   o f   s p a d e
 ( `&)   s u i t ?

 S o l u t i o n :
 L e t   B   =   B l a c k    c a r d   d r a w n
 a n d   S   =   S p a d e    c a r d   d r a w n

 S i n c e     t h e   c a r d   d r a w n  i s   b l a c k   ( e v e n t
 B   h a s     o c c u r r e d ) ,   a n d  t h e r e   a r e   1 3
 s p a d e     c a r d s   i n 26 black cards, therefore:

     EMBED Equation.3

Multiplication Law of Probability:
If there are two non-mutually exclusive events A and B such that event B
is dependent on event A, then the probability of joint event (A and B) is
given by:

P(A and B) = P(A) × P(B/A)

or
               P ( A )"B )   =    P ( A )              ×       P ( B / A )    o r
 P ( A )   +    P ( B )        P ( A U B )
     E M B E D    E q u a t i o n . 3

 o r   i t         m a y       e q u i v a l e n t l y         b e     w r i t t e n
 a s :

 P ( A )"B )         =     P ( B )    ×   P ( A / B )

 W h e n   t w o   e v e n t s            A   a n d        B     a r e
 i n d e p e n d e n t :

 P ( A )"B )         =     P ( A )    ×   P ( B )

 W h e n   t w o   e v e n t s   A   a n d   B   a r e   m u t u a l l y
 e x c l u s i v e ,   t h e i r   j o i n t   p r o b a b i l i t y
 i s   z e r o :

 P ( A )"B )         =     0

 E   x   a m p l e :
 I   n     a   g r a d u a t e   c o l l e g e ,   t h e r e   a r e   5 0 0
 m   a   l e   a n d   f e m a l e   s t u d e n t s   l e a r n i n g
 B   .   S c   a n d   B . C o m .     T h e   b r e a k   u p   i s   a s
 f   o   l l o w s :

      M a l e      F e m a l e    T o t a l          B . S c         7 0   1 5 0    2 2 0
     B . C o m      1 2 0   1 6 0    2 8 0          T o t a l         1 9 0   3 1 0    5 0 0

 W h a t   i s   t h e   p r o b a bility that a randomly selected
student is (i) a female B.Sc student, and (ii) a male B.Com student?

Solution:
a female B.Sc student:

Let A = student selected is learning B.Sc;   EMBED Equation.3
B = student selected is a female;   EMBED Equation.3



                                     A )"B   =   f e m a l e    s t u d e n t
 l e a r n i n      g   B . S c
 P ( A )"B )        =   P ( A )   ×   P ( B / A ) ,   w h e r e
 E M B E D   E      q u a t i o n . 3
           E M      B E D   E q u a t i o n . 3

 a       m a l e     B . C o m       s t u d e n t :

 L e t       E M B E D   E q u a t i o n . 3                             =
 s t u d e n t   s e l e c t e d   i s   n o t                   l e a r n i n g
 B . S c ;       E M B E D   E q u a t i o n . 3
     E M B E D   E q u a t i o n . 3                                     = student selected
is not a female;   EMBED Equation.3
  EMBED Equation.3
  EMBED Equation.3

Addition Law of Probability:
If A and B are two non-mutually exclusive events, then the probability of
the composite event (A or B) is given by:

P(A or B) = P(A) + P(B) – P(A and B)

or



                                                                                          P (
A U B )   =    P ( A )   +   P ( B )                        P ( A )"B )
     E M B E D    E q u a t i o n . 3

 F o r      t w o        m u t u a l l y        e x c l u s i v e          e v e n t s    A
 a n d      B :

 P ( A U B )         =       P ( A )    +      P ( B )

 E x a m p l e :
 A   c a r d   i s   d r a w n                 f r o m   a   p a c k   o f   5 2
 c a r d s .     W h a t   i s                 t h e   p r o b a b i l i t y
 t h a t :

 i t       i s   e i t h e r           A c e   o r   K i n g
 i t       i s   e i t h e r           Q u e e n   o r   a   D i a m o n d             ( f&)

 S o l u t i o n :
 i t   i s   e i t h e r               A c e    o r      K i n g :

 L e t      A   =   A c e ;       E M B E D   E q u a t i o n . 3
 B   =      K i n g ;       E M B E D   E q u a t i o n . 3

 ( A U B    )   =   t h        e   c a r d   d r a w n            i s    e i t h e r     a n
 A c e      o r   a   K        i n g
 P A U B    )   =   P (        A )   +   P ( B )
     E M    B E D   E q        u a t i o n . 3

 i t       i s   e i t h e r           Q u e e n      o r     a    D i a m o n d :

       L e t     C       =     Q u e e n ;            E M B E D         E q u a t i o n . 3

 D     =    D i a m o n d ;                 E M B E D       E q u a t i o n . 3

 C   U D   =   C a r d   d r a w n   i s   e i t h e r   a   Q u e e n
 o   r   a   D i a m o n d
 P   ( C U D )   =   P ( C )   +   P ( D )       P ( C )"D ) ;   w h e r e
 P   ( C )"D )   =   P ( C )   ×   P ( D / C )   =       E M B E D
    E q u a t i o n . 3
    P ( C U D )       E M B E D                       E q u a t i o n . 3

    B a y e             s       T h e o r e m :

    T   h   e       B   a y e   s           T h e o   r e m     i s   b a s e d   o n
    c   o   n   d   i   t i o n s           p r o b   a b i l   i t i e s .     I t
    c   a   l   c   u   l a t e s           p r o b   a b i l   i t i e s   o f   t h e
    c   a   u   s   e   s   t h a       t     m a y     h a v   e   p r o d u c e d   a n
    o   b   s   e   r   v e d   e       v   e n t .
    G   i   v   e   n     A 1 ,         A   2 ,   &   & & ,   A i ,   & & & & . ,   A n
    m   u   t   u   a   l l y   e       x   c l u s   i v e   e v e n t s ,   w h o s e
    u   n   i   o   n     i s   t       h   e   e n   t i r e   p o s s i b i l i t y
    s   p   a   c   e     S ,   a       n   d   l e   t   B   a n   a r b i t r a r y
    e   v   e   n   t     i n   S       ,     s u c   h   t h a t   P ( B )   `" O ,
    t   h   e   n   :

            E M B E D            E q u a t i o n . 3

    W h e r e               i    =      1 ,     2 ,     3 ,     4 ,   & & . ,   n

 P ( A          i )   a r e             c a l   l e d     p r    i o r
 p r o          b a b i l i           t i e s       a n d   P    ( A i / B )    a r   e
 c a l          l e d     p           o s t e   r i o r   p r    o b a b i l i t i    e s   .
 T h u          s   B a y e             s   T   h e o r e m      i s   a   p r o c    e s s   t o
 r e v          i s e   t h           e   p r   i o r   p r o    b abilities using    additional
sample          information           and get   the posterior    probabilities.

A           B           C         D

B           C           D         A

B                       C




A                       D

A                       B




C                       D

S

E5 =
   {TT}

S

        EMBED Equation.3
Event E5

Event   EMBED Equation.3

Venn Diagram showing the event E5 and its complementary event   EMBED
Equation.3

A and B are mutually exclusive events, as the card cannot be an Ace and a
King at the same time.

Non-Mutually Exclusive Events

Mutually Exclusive Events

A          B

S

A              B

S

C and D are not mutually exclusive events
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Eq_ 1 2 1 5 7 1 7 0 5 6                                                           '
 •   B    Î      À      F      `Xªüª´Æ
€³üª´Æ              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          V                   C o m p
O b j                                                               A   C     ÿÿÿÿ
                                      W   f           O b j I n f o
                                              ÿÿÿÿD    ÿÿÿÿ
               Y             uation
   Equation.3 ô9²q
                                                             gÍ”           8•I `ƒI

†=
     •n •u •m •b •e •r •          •o •f •
  •o •u •t •c •o •m •e          •s •    •i •n • •t •h •e •
  •e •v •e •n •t •        •A      •n •u •m •b •e •r •   •o •f •
  •o •u •t •c •o •m •e          •s •    •i •n • •t •h •e •
  •p •o •s •s •i •b •i          •l •i •t •y •   •s •p •a •c •e •
  •S    ƒP ‚( ƒA ‚) †=E   q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                 Z   °        _ 1 2 1 5
7 1 7 1 0 0                                             ÿÿÿÿÿÿÿÿG    Î
      À      F    €³üª´Æ
€³üª´Æ              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                 a               C o m p
O b j                                                   F   H    ÿÿÿÿ
                                  b   f
ƒn ‚( ƒA ‚)   ƒn ‚( ƒS ‚)     2
                                                 þÿ
ÿÿÿÿ Î      À       F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                       gÍp   ø¦I ŒŒI

ƒP ‚( ƒA ‚) †=
ƒn ‚( ƒA ‚)   ƒn ‚( ƒS ‚)   †=
  ˆ1     ˆ4†= ˆ0 ‚. ˆ2 ˆ5
                   O b j I n f o
            ÿÿÿÿI   ÿÿÿÿ                                         d           E
 q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    e   Œ             _ 1 2 1 5
7 1 2 0 2 5                                                     ÿÿÿÿÿÿÿÿL    Î
     À      F     IÄüª´Æ IÄüª´Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    h
                                                           þÿ
  ÿÿÿÿ Î          À   F   Microsoft Equation 3.0
DS Equation
Equation.3 ô9²q
                  gÍ,   ìmI èyI
    ƒm       ƒn       –(   –
)          ”
                                                            þÿ
    ÿÿÿÿ Î            À        F   Microsoft Equation 3.0
     DS
Eq    C o m p O b j                                                                                    K
 M     ÿÿÿÿ                                                         i     f                 O b j I n f o
                                                                         ÿÿÿÿN               ÿÿÿÿ
                                         k                  E q u a t i o n
  N a t i v e
     ÿÿÿÿÿÿÿÿÿÿÿÿ                                                             l     H            _ 1 2 1 5
7 1 2 0 8 1                                                                              ï     T     Q       Î
        À          F
€³üª´Æ IÄüª´Æ                            ýÿÿÿƒ       …     „     †     ˆ     ‰     °      ‹     Š     Œ     Ž
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Í      Î     Ï     Ð     Ñ     Ò     Ó     Ô     Õ     Ö      ×     Ø     Ù     Ú     Û     Ü     Ý     Þ     ß
     à     á     â     ã     ä      å     æ     ç     è     é     ê     ë     ì     í      î     ï     ð     ñ
    ò     ó     ô     õ     ö     ÷     ø     ù     þÿÿÿþÿÿÿü          þÿÿÿýÿÿÿÿ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                               n                     C o m p
O b j                                                                                   P     R      ÿÿÿÿ
                                                  o     f               O b j I n f o
                                                      ÿÿÿÿS              ÿÿÿÿ
               q                         E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                               r    p              uation
   Equation.3 ô9²q
                                                                                  gÍT
ªI 8•I
ƒP ‚( ƒA ‚) †=             +
ƒn †’!†-"
‚l ‚i ‚m
ƒm     ƒn
                        þÿ
ÿÿÿÿ Î          À   F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                    _ 1 2 1 5 7 1 2 4 2 8
                                               ÿÿÿÿÿÿÿÿV    Î      À      F
  êËüª´Æ
êËüª´Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         t                  C o m p
O b j                                                             U   W     ÿÿÿÿ
                                     u   f           O b j I n f o
                                             ÿÿÿÿX    ÿÿÿÿ
               w                E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         x     L           _ 1 2 1 5
7 1 7 3 3 0                                                       E   h      [    Î
     À       F       êËüª´Æ
êËüª´Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         z                 C o m p
O b j                                                             Z    \   ÿÿÿÿ
                                     {   f              gÍ0           ìmI èyI

ƒP ‚( ƒA ‚) †=
ƒm     ƒn
                                                 þÿ
ÿÿÿÿ Î          À   F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                       gÍh   8•I `ƒI

ƒP ‚( ƒA ‚) †=
ƒm   ƒn   †=
     ˆ6 ˆ3   ˆ3 ˆ6 ˆ0      †= ˆ0 ‚. ˆ1 ˆ7 ˆ5       O b j I n f o
                                                    ÿÿÿÿ]        ÿÿÿÿ
                         }             E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                             ~     „           _ 1 2 1 5
7 1 2 6 6 1                                                           O   w      `    Î
     À       F       êËüª´Æ
êËüª´Æ                  O l e

      ÿÿÿÿÿÿÿÿÿÿÿÿ                                        •            þÿÿÿþÿÿÿƒ
     þÿÿÿþÿÿÿþÿÿÿþÿÿÿˆ       þÿÿÿþÿÿÿ‹   þÿÿÿþÿÿÿŽ   þÿÿÿþÿÿÿþÿÿÿþÿÿÿ“    þÿÿÿþÿÿ
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     à    þÿÿÿþÿÿÿã    þÿÿÿþÿÿÿþÿÿÿþÿÿÿè     þÿÿÿþÿÿÿë    ì   þÿÿÿþÿÿÿï    þÿÿÿþÿ
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                                                     þÿ
     ÿÿÿÿ Î      À           F   Microsoft Equation 3.0
 DS Equation
 Equation.3 ô9²q
                                                     gÍ   ìmI èyI

"•A
                       þÿ
 ÿÿÿÿ Î     À      F        Microsoft Equation 3.0
     DS
Eq    C o m p O b j                                                                   _
 a     ÿÿÿÿ                                        ‚    f                  O b j I n f o
                                                           ÿÿÿÿb            ÿÿÿÿ
                                 „            E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                             …    0               _ 1 2 1 5
7 1 2 7 0 3                                                           ÿÿÿÿ†       e    Î
     À       F        êËüª´Æ
êËüª´Æ                   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                             †                      C o m p
O b j                                                                  d     f      ÿÿÿÿ
                                      ‡   f            O b j I n f o
                                              ÿÿÿÿg     ÿÿÿÿ
               ‰                 E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                             Š    X               uation
   Equation.3 ô9²q
                                                                gÍ<              À¨I d•I

ƒP ‚( 2ƒA        ‚) †= ˆ1 †"ƒP ‚( ƒA ‚)

               þÿ
     ÿÿÿÿ Î    À         F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                   _ 1 2 1 5 7 1 7 6 2 2
                                              ÿÿÿÿÿÿÿÿj    Î      À      F
  êËüª´Æ
êËüª´Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Œ                 C o m p
O b j                                                           i   k     ÿÿÿÿ
                                     •   f          O b j I n f o
                                            ÿÿÿÿl    ÿÿÿÿ
               •               E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        •    0           _ 1 2 1 5
7 1 7 6 7 0                                                     Y   •      o    Î
     À       F
êËüª´Æ à Õüª´Æ                   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ‘                C o m p
O b j                                                           n    p   ÿÿÿÿ
                                     ’   f             gÍ           ìmI èyI

2ƒA        à
                         þÿ
  ÿÿÿÿ Î        À    F        Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                          gÍ(           À¨I d•I

ƒn ‚( 2ƒA       ‚) †= ˆ1

               O b j I n f o
       ÿÿÿÿq    ÿÿÿÿ                                       ”               E q u
a t i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       •    D         _ 1 2 1 5
7 1 7 7 1 7                                                     ÿÿÿÿÿÿÿÿt    Î
     À       F    à Õüª´Æ à Õüª´Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       —
             C o m p O b j
      s   u   ÿÿÿÿ                                         ˜    f           O b j
 I n f o                                                      ÿÿÿÿv          ÿÿÿÿ
                                     š           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       ›    •            _ 1 2 1 5
7 1 3 9 9 6                                                     c   |      y    Î
     À       F     à Õüª´Æ à Õüª´Æ               þÿ
  ÿÿÿÿ Î       À       F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               gÍt   ¤qI Œ¸I

ƒP ‚( 2ƒA      ‚) †=
ƒn ‚( 2ƒA   ‚)   ƒn ‚( ƒS ‚)   †=
  ˆ1 ˆ8       •S •i •n •c •e •
 ƒP ‚( 2ƒA     ‚) †= ˆ1 †"ƒP ‚( ƒA ‚)   •o   •r   •
 ƒP ‚( ƒA ‚) †= ˆ1 †"ƒP ‚( 2ƒA     ‚)   •t   •h   •e   •r   •e   •
f •o •r •e • ƒP ‚( ƒA ‚) †= ˆ1 †"
ˆ1   ˆ8   †=
ˆ7     ˆ8   †= ˆ0 ‚. ˆ8 ˆ7 ˆ5

           þÿ
ÿÿÿÿ Î          À     F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               gÍ   T•I ¤´I

ƒP ‚( ƒB ‚/ ƒA ‚) †=
  ƒP ‚( ƒA ‹)"ƒB ‚)   ƒP ‚( ƒA ‚)      —
        •( •d •e     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ¢                C o m p
O b j                                                          x   z     ÿÿÿÿ
                                  £   f           O b j I n f o
                                          ÿÿÿÿ{    ÿÿÿÿ
               ¥             E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ¦   ¼           •p   •e   •
n •d •e •n •t •
  •e •v •e •n •t     •)
                          þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               gÍX   ÀºI   ÎI

ƒP ‚( ƒA ‚/ ƒB ‚) †=
  ƒP ‚( ƒA ‹)"ƒB ‚)    ƒP ‚( ƒB ‚)

             _ 1 2 1 5 7 1 4 0 0 4
     ÿÿÿÿå      ~    Î      À      F       à Õüª´Æ à Õüª´Æ                    O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         ©                   C o m p
O b j                                                             }    •     ÿÿÿÿ
                                       ªf            O b j I n f o
                                         ÿÿÿÿ€      ÿÿÿÿ
               ¬              E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         -    t             _ 1 2 1 5
7 1 8 3 4 5                                                       r    ‹      ƒ    Î
     À       F   à Õüª´Æ à Õüª´Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         ¯                   C o m p
O b j                                                            ‚   „     ÿÿÿÿ
                                   °   f         þÿ
  ÿÿÿÿ Î        À       F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               gÍP   ìmI èyI

ƒP ‚( ƒS ‚/ ƒB ‚) †=
ˆ1 ˆ3   ˆ2 ˆ6       †= ˆ0 ‚. ˆ5
                                                     þÿ
ÿÿÿÿ Î       À        F     Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q                                                 O b j I n f o
                                                       ÿÿÿÿ…      ÿÿÿÿ
                              ²             E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           ³    l        _ 1 2 1 5
7 1 3 7 1 2                                                        ÿÿÿÿÿÿÿÿˆ    Î
     À       F     à Õüª´Æ à Õüª´Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           µ                 C o m p
O b j                                                           ‡    ‰     ÿÿÿÿ
                                       ¶    f          O b j I n f o
                                             ÿÿÿÿŠ    ÿÿÿÿ
               ¸                  E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           ¹    ˜        _ 1 2 1 5
7 1 9 0 4 6                                                        ÿÿÿÿÿÿÿÿ•    Î
     À       F       ³Üüª´Æ   ³Üüª´Æ
                                          gÍ|              ìmI èyI
ƒP ‚( 2ƒA     ‹)"2ƒB     ‚) †= ƒP ‚(            ƒA ‹*"ƒB       ‚)    •
  •o •r •
 ˆ1 †"ƒP ‚( ƒA ‹*"ƒB ‚)
                                                                                 þÿ
  ÿÿÿÿ Î      À         F     Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                             O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                ¼                C o m p
O b j                                                Œ    Ž    ÿÿÿÿ
                            ½   f           O b j I n f o
                                    ÿÿÿÿ•    ÿÿÿÿ
              ¿        E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                À     \             gÍ@
      ìmI èyI
ƒP ‚( ƒA ‚) †=
ˆ2 ˆ2 ˆ0       ˆ5 ˆ0 ˆ0
                                                     þÿ
ÿÿÿÿ Î         À        F   Microsoft Equation 3.0
DS Equation
Equation.3 ô9²q
                  gÍ@
ªI d•I
ƒP ‚( ƒB ‚) †=
   ˆ3 ˆ1 ˆ0 ˆ5 ˆ0 ˆ0                                                        _
1 2 1 5 7 1 9 0 6 2                                                         m   ©
 ’     Î      À     F     ³Üüª´Æ     ³Üüª´Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Â                    C o m p
O b j                                                       ‘         “    ÿÿÿÿ
                                     Ã  f          O b j I n f o
                                         ÿÿÿÿ”    ÿÿÿÿ
               Å              E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Æ        \        _ 1 2 1 5
7 1 9 3 4 5                                                         ÿÿÿÿÿÿÿÿ—
    Î      À       F       ³Üüª´Æ   ³Üüª´Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        È                    C o m p
O b j                                                               –
   ˜   ÿÿÿÿ                                        É   f
                                                           þÿ
  ÿÿÿÿ Î      À        F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               gÍH   • I @ÁI

ƒP ‚( ƒB ‚/ ƒA ‚) †=
ˆ1 ˆ5 ˆ0       ˆ2 ˆ2 ˆ0
                                                     þÿ
ÿÿÿÿ Î         À        F   Microsoft Equation 3.0
   DS
Eq O b j I n f o                                                             ÿÿÿ
ÿ™   ÿÿÿÿ                                     Ë                 E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Ì     d              _ 1 2 1 5
7 1 9 4 4 9                                                     •   Ÿ      œ    Î
     À       F   ³Üüª´Æ     ³Üüª´Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Î                     C o m p
O b j                                                      ›        •     ÿÿÿÿ
                                     Ï f          O b j I n f o
                                        ÿÿÿÿž    ÿÿÿÿ
               Ñ             E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Ò     À          _ 1 2 1 5
8 6 6 8 3 6                                                      ÿÿÿÿÿÿÿÿ¡    Î
     À       F     ³Üüª´Æ   ³Üüª´Æ            uation
   Equation.3 ô9²q
                                                          gͤ             t¸I Ä¥I

†=
ˆ2 ˆ2 ˆ0   ˆ5 ˆ0 ˆ0   †×
ˆ1 ˆ5 ˆ0   ˆ2 ˆ2 ˆ0   †=
ˆ1 ˆ5 ˆ0   ˆ5 ˆ0 ˆ0   †=
ˆ3     ˆ1 ˆ0   †= ˆ0 ‚. ˆ3
                                  þÿ
ÿÿÿÿ Î         À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q                                             O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Õ                C o m p
O b j                                                               ¢    ÿÿÿÿ
                                     Ö   f          O b j I n f o
                                            ÿÿÿÿ£    ÿÿÿÿ
               Ø               E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Ù   0
                                                                åÏ          ìmI
 èyI
2ƒA
                         þÿ
  ÿÿÿÿ Î      À      F        Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                   åÏŒ          ¸¦I à¦I

ƒP ‚( 2ƒA    ‚) †= ˆ1 †"ƒP ‚( ƒA ‚) †= ˆ1 †"
  ˆ2 ˆ2 ˆ0    ˆ5 ˆ0 ˆ0   _ 1 2 1 5 8 6 6 9 1 8
                            ÿÿÿÿÿÿÿÿ¦    Î      À      F        ÀÚåüª´Æ ÀÚåüª´Æ
                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Ú                    C o m p
O b j                                                           ¥    §     ÿÿÿÿ
                                   Û   f          O b j I n f o
                                          ÿÿÿÿ¨    ÿÿÿÿ
               Ý             E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Þ     ¨             _ 1 2 1 5
8 6 6 8 8 1                                                    š   ¸      «    Î
     À       F   ÀÚåüª´Æ ÀÚåüª´Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      á                    C o m p
O b j                                                           ª    ¬     ÿÿÿÿ
                                   â   f       †=
ˆ2 ˆ8 ˆ0       ˆ5 ˆ0 ˆ0
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                                                     åÏ   DÅI   •I

"•B
                       þÿ
 ÿÿÿÿ Î     À      F        Microsoft Equation 3.0
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Eq O b j I n f o                                                            ÿÿÿ
ÿ-   ÿÿÿÿ                                    ä                 E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     å     0             _ 1 2 1 5
8 6 6 9 5 7                                                     ¤   ³      °    Î
     À       F  ÀÚåüª´Æ ÀÚåüª´Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     æ                    C o m p
O b j                                                           ¯   ±     ÿÿÿÿ
                                   ç   f         O b j I n f o
                                         ÿÿÿÿ²    ÿÿÿÿ
               é            E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     ê     ¨          _ 1 2 1 5
8 6 7 0 1 1                                                     ÿÿÿÿÿÿÿÿµ    Î
     À       F   ÀÚåüª´Æ ÀÚåüª´Æ             uation
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                                                         åÏŒ            ØnI hÛI

ƒP ‚( 2ƒB      ‚) †= ˆ1 †"ƒP ‚( ƒB ‚) †= ˆ1 †"
ˆ3 ˆ1 ˆ0   ˆ5 ˆ0 ˆ0   †=
ˆ1 ˆ9 ˆ0       ˆ5 ˆ0 ˆ0
                                                     þÿ
ÿÿÿÿ Î         À        F   Microsoft Equation 3.0
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   Equation.3 ô9²q                                                              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                       í                   C o m p
O b j                                                                           ´     ¶    ÿÿÿÿ
                                                  î    f         O b j I n f o
                                                      ÿÿÿÿ·       ÿÿÿÿ
               ð                         E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                       ñ    ø
                                                                                    åÏÜ       ОI
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2ƒA        ‹)"2ƒB            †=      •    •s   •t   •u   •d   •e   •n   •t   •
  •s   •e    •l •e    •c        •t      •e   •d   •    •i   •s   •
  •l   •e    •a •r    •n        •i      •n   •g   •    •B   •.   •C   •o   •m   •
  •a   •n    •d •     •a        •
  •m   •a    •l •e         2
                                   þÿ
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   Equation.3 ô9²q                                         _ 1 2 1 5 8 6 7 1
 2 5                                               ®   Ñ      º    Î      À
     F    ÀÚåüª´Æ ÀÚåüª´Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    õ                 C o m p
O b j                                                       ¹    »    ÿÿÿÿ
                                ö   f           O b j I n f o
                                        ÿÿÿÿ¼    ÿÿÿÿ
              ø
                         åÏX      …I , J
ƒP ‚( 2ƒA      ‹)"2ƒB    ‚) †= ƒP ‚( 2ƒA        ‚) †× ƒP ‚( 2
ƒB     ‚/ 2ƒA      ‚) •, •    •w •h •e •r •e •
 ƒP ‚( 2ƒB      ‚/ 2ƒA    ‚) †=
ˆ1 ˆ2 ˆ0   ˆ2 ˆ8 ˆ0   —   †=
ˆ2 ˆ8 ˆ0   ˆ5 ˆ0 ˆ0   †×
ˆ1 ˆ2 ˆ0   ˆ2 ˆ8 ˆ0   †=
  ˆ1 ˆ2 ˆ0    ˆ5 ˆ0 ˆ0   †= ˆ0 ‚. ˆ2 ˆ4
                                                               E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ù   t        _ 1 2 1 5
8 6 8 6 4 9                                                    à   ÿÿÿÿ¿    Î
     À      F     à{íüª´Æ à{íüª´Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ÿ               C o m p
O b j                                                          ¾    À   ÿÿÿÿ
                                        f            þÿÿÿþÿÿÿ      þÿÿÿþÿÿÿ
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    þÿÿÿþÿÿÿ
    þÿÿÿþÿÿÿ     þÿÿÿþÿÿÿ     þÿÿÿþÿÿÿ        þÿÿÿþÿÿÿ     þÿÿÿþÿÿÿ      þÿÿÿþÿ
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ÿÿþÿÿÿ4    þÿÿÿþÿÿÿ7    8   9    :   þÿÿÿþÿÿÿ=    þÿÿÿþÿÿÿ@    þÿÿÿþÿÿÿC    þÿÿ
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ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ þÿ
   ÿÿÿÿ Î       À      F    Microsoft Equation 3.0
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                                                   åÏT   8•I `ƒI

ƒP ‚( 2ƒA     ‹*"2ƒB      ‚) †=   •
 ˆ1 †"ƒP ‚( ƒA ‹)"ƒB ‚)
                                              þÿ
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   Equation.3 ô9²q                                           O b j I n f o
                                                   ÿÿÿÿÁ      ÿÿÿÿ
                                      E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          p        _ 1 2 1 5
8 6 7 7 6 8                                                  ÿÿÿÿÿÿÿÿÄ    Î
     À       F    à{íüª´Æ à{íüª´Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                      C o m p
O b j                                                        Ã   Å    ÿÿÿÿ
                                     f           O b j I n f o
                                         ÿÿÿÿÆ    ÿÿÿÿ
                            E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ
         P       _ 1 2 1 5 8 6 7 8 1 3
               Ì   É    Î      À      F     à{íüª´Æ à{íüª´Æ
                                                                        åÏ4
      ìmI èyI
ƒP ‚( ƒA ‚) †=
ˆ4     ˆ5 ˆ2
                                                 þÿ
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   Equation.3 ô9²q
                       åÏ4   \¦I ToI

ƒP ‚( ƒB ‚) †=
ˆ4   O l e
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           C o m p O b j
    È   Ê    ÿÿÿÿ
f           O b j I n f o
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          E q u a t i o n
N a t i v e
  ÿÿÿÿÿÿÿÿÿÿÿÿ                                    P   ˆ5 ˆ2

                                             þÿ
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  Equation.3 ô9²q
                    åψ   ìmI èyI

†=
ˆ4   ˆ5 ˆ2   †+
ˆ4   ˆ5 ˆ2   †=
ˆ8   ˆ5 ˆ2   †=
  ˆ2 ˆ1 ˆ3    †= ˆ0 ‚. ˆ1 ˆ5 ˆ4
   _ 1 2 1 5 8 6 8 0 0 6                                                       ÿÿ
ÿÿÿÿÿÿÎ    Î      À      F    à{íüª´Æ à{íüª´Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                           C o m p
O b j                                                         Í     Ï      ÿÿÿÿ
                                       f          O b j I n f o
                                          ÿÿÿÿÐ    ÿÿÿÿ
                             E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           ¤             _ 1 2 1 5
8 6 8 1 6 3                                                   Ç     Û      Ó    Î
     À       F    à{íüª´Æ   £öüª´Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                           C o m p
O b j                                                         Ò     Ô      ÿÿÿÿ
                                       f
                                        þÿ
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                       åÏ4   ìmI èyI

ƒP ‚( ƒC ‚) †=
ˆ4     ˆ5 ˆ2
                                                 þÿ
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Eq O b j I n f o                                                    ÿÿÿ
ÿÕ   ÿÿÿÿ                                              E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   P          _ 1 2 1 5
8 6 8 1 9 2                                             ÿÿÿÿÿÿÿÿØ    Î
     À       F   £öüª´Æ   £öüª´Æ         O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                              -                   C o m p
O b j                                                   ×   Ù     ÿÿÿÿ
                                    -
   f          O b j I n f o
       ÿÿÿÿÚ   ÿÿÿÿ                               !               E q u a
 t i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                              "    T             _ 1 2 1 5
8 6 8 3 7 2                                             Ö   ½      Ý    Î
     À       F      £öüª´Æ £öüª´Æ       uation
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                                                 åÏ8            Œ¦I ToI

ƒP ‚( ƒD ‚) †=
ˆ1 ˆ3   ˆ5 ˆ2
                                                   þÿ
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                                           O l e

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O b j                                                  Ü   Þ    ÿÿÿÿ
                          %   f           O b j I n f o
                                  ÿÿÿÿß    ÿÿÿÿ
               '     E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                              (     h              åÏL
      ìmI èyI
ˆ4   ˆ5 ˆ2   †×
ˆ1   ˆ4   †=
ˆ1     ˆ5 ˆ2
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                    åϨ   \¦I ToI

†=
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  ˆ1 _ 1 2 1 5 8 6 8 4 4 3
   ÿÿÿÿÿÿÿÿâ   Î      À      F           £öüª´Æ   £öüª´Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                            *                 C o m p
O b j                                                              á    ã     ÿÿÿÿ
                                     +    f         O b j I n f o
                                          ÿÿÿÿä      ÿÿÿÿ
               -             E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                            .   Ä        _ 1 2 1 5
7 1 5 4 1 0                                                        ÿÿÿÿÿÿÿÿç    Î
     À       F     £öüª´Æ   £öüª´Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                            2                 C o m p
O b j                                                              æ     è    ÿÿÿÿ
                                     3    f       ˆ6   ˆ5 ˆ2        †=
ˆ4     ˆ1 ˆ3   †= ˆ0 ‚. ˆ3 ˆ0 ˆ8

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ƒP ‚( ƒA
ƒi
‚/ ƒB ‚) †=
  ƒP ‚( ƒB ‹)"ƒA
ƒi
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 i o n    N a t i v e
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6 9 8 6 6 1                                          ÿÿÿÿÿÿÿÿì    Î
      À      F     £öüª´Æ   £öüª´Æ   O l e

     ÿÿÿÿÿÿÿÿÿÿÿÿ                       ;                 ‚( ƒB
‚)        —   †=
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ƒi
‚) †× ƒP ‚( ƒA ‚)      ƒP ‚( ƒB ‚/ ƒA
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ƒi
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DOCUMENT INFO
Description: bank reconcilation statement,circular flow of income, closing enteries, coefficient of correlation, concepts of National income, Consumption Functions, Determination of National income,Financial statement, Flow of Funds accounts, Index numbers, Introduction to statistics,investment, macro economics, measures of central tendency, measure of depersion, measure of shape distribution, merchandise inventory, micro macro economics, Micro Economics, multiplier and accelerator, probability, public finance, random variable,regression,sample distribution theory, statistics dynamics and comparative, theofy of employment, trade cycle