Use the moment-area theorems and determine the slope at B and deflection at B. EI is constant.

1. Page 1 of 18
CHAPTER ONE
INTRODUCTION
1.1. Deterministic and non-deterministic models
1.2. Review of set theory: sets, union, intersection, complementation, De-Morgan’s rules
1.3. Random experiments, sample space and events
1.4. Finite sample spaces and equally likely outcomes
1.5. Counting techniques
1.6. Definitions of probability
1.7. Derived theorems of probability
1.1. Deterministic and non-deterministic models
Models are integral parts of both social and natural sciences. In both cases we construct or fit
models to represent the interrelationship between two or more variables. Particularly, in the
fields like Statistics and Economics models are fitted for the sake of forecasting. I
It
t i
is
s p
po
os
ss
si
ib
bl
le
e t
to
o
c
cl
la
as
ss
si
if
fy
y m
mo
od
de
el
ls
s i
in
n t
to
o d
di
if
ff
fe
er
re
en
nt
t g
gr
ro
ou
up
ps
s b
ba
as
se
ed
d o
on
n v
va
ar
ri
ie
ed
d a
at
tt
tr
ri
ib
bu
ut
te
es
s o
or
r c
cr
ri
it
te
er
ri
ia
a.
. B
Ba
as
se
ed
d o
on
n t
th
he
e t
ty
yp
pe
e o
of
f
e
ex
xp
pe
er
ri
im
me
en
nt
t f
fo
or
r w
wh
hi
ic
ch
h w
we
e f
fi
it
t t
th
he
e m
mo
od
de
el
l,
, w
we
e c
cl
la
as
ss
si
if
fy
y m
mo
od
de
el
l a
as
s D
De
et
te
er
rm
mi
in
ni
is
st
ti
ic
c a
an
nd
d N
No
on
n-
-
d
de
et
te
er
rm
mi
in
ni
is
st
ti
ic
c m
mo
od
de
el
ls
s.
.
Definition: Experiment is any activity (process or action) that we intended to do under certain
condition to obtain a well defined results, usually called the outcome of an experiment. The
possible results of an experiment may be one or more. Based on the number of possible results,
we may classify an experiment as Deterministic and Non-deterministic experiment.
A- Deterministic experiment: this is an experim ent for which the outcomes can be
predicted in advance and is known prior to its conduct. For this type
of experiment we have only one possible result (certain and unique). The result of an
experiment is assumed to be dependent on the condition under which an experiment is
performed. A mathematical form of equations to be defined on this experiment is called a
deterministic model.
Definition: Deterministic model is a model that stipulates the condition under which an
experiment is performed determine the outcome of that experiment.

2. Introduction to probability theory (Stat 276) Department of Statistics (AAU)
Page 2 of 18
Examples: An experiment conducted to verify the Newton Laws of Motion: F = ma; an
experiment conducted to determine the economic law of demand: where Q is a
quantity demand, P is the price and t is a time; etc…
Note: Most of the experiments that we conduct to verify science law is an example of
deterministic experiment.
B- Non-deterministic experiment: this is an experiment for which the outcome of a given trial
cannot be predicted in advance prior to its conduct. We also call this experiment as unpredictable
or probabilistic or stochastic or random experiment. Usually the result of this experiment is
subjected to chance and is possibly more than one. In this experiment, whatever the condition
under which an experiment is performed, one cannot tell with certainty which outcome occurs
at any particular execution of an experiment though it is possible to list those outcoes.
This types of experiments are characterized by the following three properties:
I. The experiment is repeatable under identical conditions.
II. The outcome in any particular trial is variable, i.e., it depends on some chance or random
mechanism.
III. If the experiment is repeated a large number of times, then some regularity becomes
apparent in the outcomes obtained. This regularity enables us to set some mathematical
form of equations called non-deterministic model.
Definition: Non-deterministic model is a mathematical description of an uncertain situation.
Example: The experiment of tossing a coin: there are two possible outcomes i.e. getting a
Head or Tail. Though we know the possible outcomes, we cannot for sure predict that we
can get for example a Tail if we flip the coin in a particular manner and so on.
Note: the quantitative measure of uncertainty regarding one or more outcomes of a random
experiment is called a probability.
Exercise: I
Id
de
en
nt
ti
if
fy
y t
th
he
e f
fo
ol
ll
lo
ow
wi
in
ng
g m
mo
od
de
el
ls
s a
as
s d
de
et
te
er
rm
mi
in
ni
is
st
ti
ic
c o
or
r N
No
on
n-
-d
de
et
te
er
rm
mi
in
ni
is
st
ti
ic
c
a
a)
) E
Eq
qu
ua
at
ti
io
on
n o
of
f a
a s
st
tr
ra
ai
ig
gh
ht
t l
li
in
ne
e.
.
b
b)
) V
Va
ar
ri
ia
ab
bl
le
e c
co
os
st
t=
=(
(C
Co
os
st
t p
pe
er
r u
un
ni
it
t)
)(
(Q
Qu
ua
an
nt
ti
it
ty
y p
pr
ro
od
du
uc
ce
ed
d)
)

3. Page 3 of 18
c
c)
) Y
Yi
i=
=A
AX
Xi
i+
+e
ei
i w
wh
he
er
re
e e
ei
i i
is
s t
th
he
e e
er
rr
ro
or
r t
te
er
rm
m (
(o
or
r r
ra
an
nd
do
om
m c
co
om
mp
po
on
ne
en
nt
t)
) a
as
ss
so
oc
ci
ia
at
te
ed
d w
wi
it
th
h t
th
he
e i
it
th
h
o
ob
bs
se
er
rv
va
at
ti
io
on
n.
.
1.2. Review of set theory: sets, union, intersection, complementation, De-Morgan’s rules
D
De
ef
fi
in
ni
it
ti
io
on
n a
an
nd
d T
Ty
yp
pe
es
s o
of
f S
Se
et
ts
s
S
Se
et
t:
: A
A s
se
et
t i
is
s a
a w
we
el
ll
l-
-d
de
ef
fi
in
ne
ed
d c
co
ol
ll
le
ec
ct
ti
io
on
n o
or
r l
li
is
st
t o
of
f o
ob
bj
je
ec
ct
ts
s.
.
A
An
n o
ob
bj
je
ec
ct
t t
th
ha
at
t b
be
el
lo
on
ng
gs
s t
to
o a
a p
pa
ar
rt
ti
ic
cu
ul
la
ar
r s
se
et
t i
is
s c
ca
al
ll
le
ed
d a
an
n e
el
le
em
me
en
nt
t.
. S
Se
et
ts
s a
ar
re
e u
us
su
ua
al
ll
ly
y d
de
en
no
ot
te
ed
d b
by
y c
ca
ap
pi
it
ta
al
l
l
le
et
tt
te
er
rs
s (
(A
A,
, B
B,
, C
C e
et
tc
c)
).
. O
On
n t
th
he
e o
ot
th
he
er
r h
ha
an
nd
d,
, e
el
le
em
me
en
nt
ts
s o
of
f s
se
et
ts
s a
ar
re
e u
us
su
ua
al
ll
ly
y d
de
en
no
ot
te
ed
d b
by
y s
sm
ma
al
ll
l l
le
et
tt
te
er
rs
s (
(a
a,
,
b
b,
, c
c e
et
tc
c)
).
.
E
Ex
xa
am
mp
pl
le
es
s:
: T
Th
he
e s
se
et
t o
of
f s
st
tu
ud
de
en
nt
ts
s i
in
n a
a c
cl
la
as
ss
s;
; t
th
he
e s
se
et
t o
of
f e
ev
ve
en
n n
nu
um
mb
be
er
rs
s;
; t
th
he
e s
se
et
t o
of
f p
po
os
ss
si
ib
bl
le
e o
ou
ut
tc
co
om
me
es
s
o
of
f a
an
n e
ex
xp
pe
er
ri
im
me
en
nt
t;
; e
et
tc
c.
.
N
No
ot
te
e:
: I
If
f X
X b
be
el
lo
on
ng
gs
s t
to
o s
se
et
t A
A w
we
e w
wr
ri
it
te
e X
X ∈
∈ A
A,
, a
an
nd
d i
if
f X
X d
do
oe
es
s n
no
ot
t b
be
el
lo
on
ng
g t
to
o s
se
et
t A
A w
we
e w
wr
ri
it
te
e X
X ∉
∉ A
A
T
Tw
wo
o S
Sp
pe
ec
ci
ia
al
l S
Se
et
ts
s
U
Un
ni
iv
ve
er
rs
sa
al
l s
se
et
t (
(U
U)
):
: U
Un
ni
iv
ve
er
rs
sa
al
l s
se
et
t i
is
s t
th
he
e c
co
ol
ll
le
ec
ct
ti
io
on
n o
of
f a
al
ll
l o
ob
bj
je
ec
ct
ts
s u
un
nd
de
er
r c
co
on
ns
si
id
de
er
ra
at
ti
io
on
n.
. U
Un
ni
iv
ve
er
rs
sa
al
l s
se
et
t
f
fo
or
r a
a g
gi
iv
ve
en
n d
di
is
sc
cu
us
ss
si
io
on
n i
is
s f
fi
ix
xe
ed
d a
an
nd
d p
pr
re
e-
-d
de
et
te
er
rm
mi
in
ne
ed
d.
.
E
Ex
xa
am
mp
pl
le
e:
: *
* T
Th
he
e s
se
et
t o
of
f r
re
ea
al
l n
nu
um
mb
be
er
rs
s c
ca
an
n b
be
e s
se
ee
en
n t
to
o b
be
e u
un
ni
iv
ve
er
rs
sa
al
l s
se
et
t o
of
f n
nu
um
mb
be
er
rs
s.
.
*
* C
Co
on
ns
si
id
de
er
r a
an
n e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f r
ro
ol
ll
li
in
ng
g a
a b
ba
al
la
an
nc
ce
ed
d d
di
ie
e.
. T
Th
he
en
n,
, t
th
he
e s
se
et
t o
of
f a
al
ll
l p
po
os
ss
si
ib
bl
le
e
o
ou
ut
tc
co
om
me
es
s o
of
f t
th
hi
is
s e
ex
xp
pe
er
ri
im
me
en
nt
t c
ca
an
n b
be
e c
co
on
ns
si
id
de
er
re
ed
d a
as
s t
th
he
e u
un
ni
iv
ve
er
rs
sa
al
l s
se
et
t.
. U
U=
={
{1
1,
,2
2,
,3
3,
,4
4,
,5
5,
,6
6}
}
E
Em
mp
pt
ty
y s
se
et
t (
(φ
φ o
or
r{
{}
})
):
: A
A s
se
et
t w
wi
it
th
h n
no
o e
el
le
em
me
en
nt
t i
is
s c
ca
al
ll
le
ed
d e
em
mp
pt
ty
y s
se
et
t.
.
E
Ex
xa
am
mp
pl
le
e:
: I
In
n t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f t
to
os
ss
si
in
ng
g t
tw
wo
o d
di
ic
ce
e a
at
t t
th
he
e s
sa
am
me
e t
ti
im
me
e,
, d
de
ef
fi
in
ne
e A
A t
to
o b
be
e a
a s
se
et
t w
wh
ho
os
se
e
e
el
le
em
me
en
nt
ts
s a
ar
re
e p
pa
ai
ir
rs
s o
of
f o
ou
ut
tc
co
om
me
es
s f
fr
ro
om
m t
th
he
e t
tw
wo
o d
di
ic
ce
e w
wi
it
th
h s
su
um
m g
gr
re
ea
at
te
er
r t
th
ha
an
n 2
20
0.
. O
Ob
bv
vi
io
ou
us
sl
ly
y,
, t
th
he
e s
se
et
t A
A
i
is
s e
em
mp
pt
ty
y s
se
et
t b
be
ec
ca
au
us
se
e t
th
he
e m
ma
ax
xi
im
mu
um
m p
po
os
ss
si
ib
bl
le
e s
su
um
m i
is
s 1
12
2,
, w
wh
hi
ic
ch
h i
is
s a
at
tt
ta
ai
in
ne
ed
d w
wh
he
en
n b
bo
ot
th
h t
th
he
e o
ou
ut
tc
co
om
me
es
s
a
ar
re
e 6
6.
.
S
Su
ub
bs
se
et
t:
: A
A ⊆
⊆ B
B i
if
ff
f X
X ∈
∈ A
A ⇒
⇒ X
X ∈
∈ B
B f
fo
or
r a
al
ll
l X
X e
el
le
em
me
en
nt
t i
in
n t
th
he
e u
un
ni
iv
ve
er
rs
sa
al
l s
se
et
t.
.
1
1.
.2
2.
.1
1 S
Se
et
t o
op
pe
er
ra
at
ti
io
on
ns
s
O
On
ne
e c
ca
an
n i
id
de
en
nt
ti
if
fy
y t
th
hr
re
ee
e b
ba
as
si
ic
c s
se
et
t o
op
pe
er
ra
at
ti
io
on
ns
s.
. T
Th
he
es
se
e a
ar
re
e:
:

4. Page 4 of 18
a
a.
. C
Co
om
mp
pl
le
em
me
en
nt
t:
: F
Fo
or
r a
an
ny
y s
se
et
t A
A,
, t
th
he
e c
co
om
mp
pl
le
em
me
en
nt
t o
of
f A
A d
de
en
no
ot
te
ed
d b
by
y A
A/
/
,
, o
or
r A
Ac
c
o
or
r Ā
Ā i
is
s g
gi
iv
ve
en
n b
by
y
{
{X
X ∈
∈ U
U/
/ X
X∉
∉A
A}
}
N
No
ot
te
e:
: U
U/
/
=
= φ
φ ;
; φ
φ/
/
=
= U
U ;
; (
( A
A/
/
)
)/
/
=
= A
A
b
b.
. U
Un
ni
io
on
n:
: G
Gi
iv
ve
en
n t
tw
wo
o s
se
et
ts
s A
A a
an
nd
d B
B,
, t
th
he
e u
un
ni
io
on
n o
of
f A
A a
an
nd
d B
B d
de
en
no
ot
te
ed
d b
by
y A
A∪ B
B i
is
s t
th
he
e s
se
et
t o
of
f a
al
ll
l
e
el
le
em
me
en
nt
ts
s,
, w
wh
hi
ic
ch
h b
be
el
lo
on
ng
g t
to
o s
se
et
t A
A o
or
r B
B,
, o
or
r b
bo
ot
th
h.
.
A
A∪
∪B
B =
={
{ X
X:
: X
X ∈
∈ A
A ∨
∨ X
X∈
∈ B
B}
};
; h
he
er
re
e t
th
he
e l
lo
og
gi
ic
ca
al
l c
co
on
nn
ne
ec
ct
ti
iv
ve
e ∨
∨ (
(O
OR
R)
) i
is
s u
us
se
ed
d i
in
n i
it
ts
s i
in
nc
cl
lu
us
si
iv
ve
e s
se
en
ns
se
e.
.
I
In
n g
ge
en
ne
er
ra
al
l,
, i
if
f w
we
e h
ha
av
ve
e a
a s
se
eq
qu
ue
en
nc
ce
e o
of
f s
se
et
ts
s A
A1
1,
, A
A2
2,
, …
…,
, A
An
n t
th
he
en
n
A
A1
1 ∪
∪A
A2
2∪
∪ …
… ∪
∪A
An
n =
= n}
and
1
between
i
one
least
at
for
:
{
1
i
n
i
i A
X
U
X
A ∈
∈
=
=
U
R
Re
ec
ca
al
ll
l t
th
ha
at
t A
A ∪
∪ φ
φ =
= A
A a
an
nd
d A
A ∪
∪ U
U =
= U
U
c
c.
. I
In
nt
te
er
rs
se
ec
ct
ti
io
on
n:
: F
Fo
or
r a
an
ny
y t
tw
wo
o s
se
et
ts
s A
A a
an
nd
d B
B t
th
he
e i
in
nt
te
er
rs
se
ec
ct
ti
io
on
n o
of
f A
A a
an
nd
d B
B i
is
s d
de
ef
fi
in
ne
ed
d t
to
o b
be
e t
th
he
e s
se
et
t o
of
f
a
al
ll
l e
el
le
em
me
en
nt
ts
s t
th
ha
at
t o
oc
cc
cu
ur
r i
in
n b
bo
ot
th
h s
se
et
t A
A a
an
nd
d a
al
ls
so
o s
se
et
t B
B.
. S
Sy
ym
mb
bo
ol
li
ic
ca
al
ll
ly
y,
, w
we
e w
wr
ri
it
te
e
A
A ∩
∩ B
B =
={
{ X
X ∈
∈ U
U /
/ X
X ∈
∈ A
A ∧
∧ X
X∈
∈B
B}
}
A
A1
1 ∩
∩ A
A2
2 ∩
∩ …
… ∩
∩ A
An
n =
= n}
and
1
between
i
all
for
:
{
1
i
n
i
i A
X
U
X
A ∈
∈
=
=
I
S
Se
et
ts
s w
wi
it
th
h n
no
o i
in
nt
te
er
rs
se
ec
ct
ti
io
on
n a
ar
re
e c
ca
al
ll
le
ed
d d
di
is
sj
jo
oi
in
nt
t s
se
et
ts
s.
. A
A a
an
nd
d B
B a
ar
re
e d
di
is
sj
jo
oi
in
nt
t i
if
f A
A ∩
∩ B
B =
= φ
φ.
.
1
1.
.2
2.
.2
2 P
Pr
ro
op
pe
er
rt
ti
ie
es
s o
of
f s
se
et
t o
op
pe
er
ra
at
ti
io
on
ns
s
I
I.
. C
Co
om
mm
mu
ut
ta
at
ti
iv
ve
e l
la
aw
w
i
i)
) A
A∪
∪ B
B =
= B
B∪
∪ A
A,
, a
an
nd
d
i
ii
i)
)A
A ∩
∩ B
B =
= B
B ∩
∩ A
A
I
II
I.
. A
As
ss
so
oc
ci
ia
at
ti
iv
ve
e l
la
aw
w
i
i)
) A
A∪
∪ (
(B
B ∪
∪ C
C)
) =
= (
(A
A ∪
∪ B
B)
) ∪
∪ C
C,
, a
an
nd
d
i
ii
i)
) A
A ∩
∩ (
(B
B ∩
∩ C
C)
) =
= (
(A
A ∩
∩ B
B)
) ∩
∩ C
C

5. Page 5 of 18
I
II
II
I.
.D
Di
is
st
tr
ri
ib
bu
ut
ti
iv
ve
e l
la
aw
w
i
i)
) A
A ∪
∪ (
(B
B ∩
∩ C
C)
) =
= (
(A
A ∪
∪ B
B)
) ∩
∩ (
(A
A ∪
∪ C
C)
),
,
i
ii
i)
)A
A ∩
∩ (
(B
B ∪
∪ C
C)
) =
= (
(A
A ∩
∩B
B)
) ∪
∪ (
(A
A ∩
∩ C
C)
),
,
M
Mo
or
re
e g
ge
en
ne
er
ra
al
ll
ly
y,
,
( )
( )
I U
U I
U I
I U
n
i
i
n
i
i
n
i
i
n
i
i
B
A
B
A
and
B
A
B
A
1
1
1
1
,
=
=
=
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
I
IV
V.
. D
De
em
mo
or
rg
ga
an
n’
’s
s l
la
aw
w
(
(A
A∪
∪ B
B)
)/
/
=
= A
A/
/
∩
∩ B
B/
/
a
an
nd
d (
(A
A ∩
∩B
B)
)/
/
=
= A
A/
/
∪
∪ B
B/
/
M
Mo
or
re
e g
ge
en
ne
er
ra
al
ll
ly
y,
,
U
I
I
U
n
i
i
n
i
i
n
i
i
n
i
i A
A
and
A
A
1
'
'
1
1
'
'
1
,
=
=
=
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
1.3. Random experiments, sample space and events
I
In
nt
tr
ro
od
du
uc
ct
ti
io
on
n
P
Pe
er
rh
ha
ap
ps
s t
th
he
e c
co
on
nc
ce
ep
pt
t o
of
f s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e a
an
nd
d e
ev
ve
en
nt
ts
s m
ma
ay
y b
be
e c
co
on
ns
si
id
de
er
re
ed
d a
as
s t
th
he
e c
co
or
rn
ne
er
r s
st
to
on
ne
es
s f
fo
or
r t
th
he
e
w
wh
ho
ol
le
e d
di
is
sc
cu
us
ss
si
io
on
n i
in
n p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y.
. T
Th
he
e c
co
on
nc
ce
ep
pt
t o
of
f p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y c
ca
an
n n
ne
ev
ve
er
r b
be
e a
ad
dd
dr
re
es
ss
se
ed
d w
wi
it
th
ho
ou
ut
t
f
fu
ur
rn
ni
is
sh
hi
in
ng
g t
th
he
e i
id
de
ea
a o
of
f s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e a
an
nd
d e
ev
ve
en
nt
ts
s.
. T
Th
he
e g
go
oo
od
d n
ne
ew
ws
s i
is
s t
th
ha
at
t o
on
ne
e c
ca
an
n e
ea
as
si
il
ly
y g
gr
ra
as
sp
p t
th
he
e
c
co
on
nc
ce
ep
pt
t o
of
f s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e a
an
nd
d e
ev
ve
en
nt
ts
s p
pr
ro
ov
vi
id
de
ed
d t
th
he
er
re
e i
is
s a
a c
cl
le
ea
ar
r u
un
nd
de
er
rs
st
ta
an
nd
di
in
ng
g o
of
f u
un
ni
iv
ve
er
rs
sa
al
l s
se
et
t a
an
nd
d
s
su
ub
bs
se
et
ts
s,
, w
wh
hi
ic
ch
h a
ar
re
e e
el
le
em
me
en
nt
ta
ar
ry
y c
co
on
nc
ce
ep
pt
ts
s i
in
n s
se
et
t t
th
he
eo
or
ry
y.
. W
We
e h
ha
av
ve
e a
a p
pe
er
rf
fe
ec
ct
t a
an
na
al
lo
og
gy
y.
. T
Th
he
e c
co
on
nc
ce
ep
pt
ts
s o
of
f
u
un
ni
iv
ve
er
rs
sa
al
l s
se
et
t a
an
nd
d s
su
ub
bs
se
et
ts
s i
in
n t
th
he
e t
th
he
eo
or
ry
y o
of
f s
se
et
ts
s d
di
ir
re
ec
ct
tl
ly
y c
co
or
rr
re
es
sp
po
on
nd
d,
, r
re
es
sp
pe
ec
ct
ti
iv
ve
el
ly
y,
, t
to
o t
th
he
e c
co
on
nc
ce
ep
pt
ts
s
s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e a
an
nd
d e
ev
ve
en
nt
ts
s w
wh
he
en
n i
it
t c
co
om
me
es
s t
to
o t
th
he
e t
th
he
eo
or
ry
y o
of
f p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y.
.
1
1.
. R
Ra
an
nd
do
om
m E
Ex
xp
pe
er
ri
im
me
en
nt
t:
: I
It
t i
is
s a
an
n e
ex
xp
pe
er
ri
im
me
en
nt
t t
th
ha
at
t c
ca
an
n b
be
e r
re
ep
pe
ea
at
te
ed
d a
an
ny
y n
nu
um
mb
be
er
r o
of
f t
ti
im
me
es
s u
un
nd
de
er
r
s
si
im
mi
il
la
ar
r c
co
on
nd
di
it
ti
io
on
ns
s a
an
nd
d i
it
t i
is
s p
po
os
ss
si
ib
bl
le
e t
to
o e
en
nu
um
me
er
ra
at
te
e t
th
he
e t
to
ot
ta
al
l n
nu
um
mb
be
er
r o
of
f o
ou
ut
tc
co
om
me
es
s w
wi
it
th
h o
ou
ut
t
p
pr
re
ed
di
ic
ct
ti
in
ng
g a
an
n i
in
nd
di
iv
vi
id
du
ua
al
l o
ou
ut
t c
co
om
me
e.
. R
Ra
an
nd
do
om
m e
ex
xp
pe
er
ri
im
me
en
nt
ts
s a
ar
re
e a
as
ss
so
oc
ci
ia
at
te
ed
d w
wi
it
th
h p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y
m
mo
od
de
el
ls
s.
. A
As
s t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t i
is
s r
re
ep
pe
ea
at
te
ed
d a
a l
la
ar
rg
ge
e n
nu
um
mb
be
er
r o
of
f t
ti
im
me
e a
a c
ce
er
rt
ta
ai
in
n p
pa
at
tt
te
er
rn
ns
s a
ap
pp
pe
ea
ar
r.
.

6. Page 6 of 18
S
So
om
me
e f
fe
ea
at
tu
ur
re
es
s o
of
f a
a r
ra
an
nd
do
om
m e
ex
xp
pe
er
ri
im
me
en
nt
t
9
9 E
Ea
ac
ch
h e
ex
xp
pe
er
ri
im
me
en
nt
t i
is
s c
ca
ap
pa
ab
bl
le
e o
of
f b
be
ei
in
ng
g r
re
ep
pe
ea
at
te
ed
d i
in
nd
de
ef
fi
in
ni
it
te
el
ly
y u
un
nd
de
er
r e
es
ss
se
en
nt
ti
ia
al
ll
ly
y u
un
nc
ch
ha
an
ng
ge
ed
d
c
co
on
nd
di
it
ti
io
on
ns
s,
,
9
9 A
Al
lt
th
ho
ou
ug
gh
h w
we
e a
ar
re
e,
, i
in
n g
ge
en
ne
er
ra
al
l,
, u
un
na
ab
bl
le
e t
to
o d
de
et
te
er
rm
mi
in
ne
e w
wh
hi
ic
ch
h p
po
os
ss
si
ib
bl
le
e o
ou
ut
tc
co
om
me
e w
wi
il
ll
l r
re
es
su
ul
lt
t i
in
n a
a
s
si
in
ng
gl
le
e t
tr
ri
ia
al
l o
of
f t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t,
, w
we
e a
ar
re
e a
ab
bl
le
e t
to
o d
de
es
sc
cr
ri
ib
be
e t
th
he
e s
se
et
t o
of
f a
al
ll
l p
po
os
ss
si
ib
bl
le
e o
ou
ut
tc
co
om
me
es
s.
.
9
9 A
As
s t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t i
is
s r
re
ep
pe
ea
at
te
ed
d a
a l
la
ar
rg
ge
e n
nu
um
mb
be
er
r o
of
f t
ti
im
me
es
s a
a d
de
ef
fi
in
ni
it
te
e p
pa
at
tt
te
er
rn
n o
or
r r
re
eg
gu
ul
la
ar
ri
it
ty
y i
is
s
e
ex
xp
pe
ec
ct
te
ed
d t
to
o a
ap
pp
pe
ea
ar
r.
. F
Fr
ro
om
m t
th
hi
is
s r
re
eg
gu
ul
la
ar
ri
it
ty
y w
we
e c
ca
an
n c
co
on
ns
st
tr
ru
uc
ct
t a
a p
pr
re
ec
ci
is
se
e m
ma
at
th
he
em
ma
at
ti
ic
ca
al
l m
mo
od
de
el
l.
.
E
Ex
xa
am
mp
pl
le
e:
: C
Co
on
ns
si
id
de
er
r t
th
he
e r
ra
an
nd
do
om
m e
ex
xp
pe
er
ri
im
me
en
nt
ts
s g
gi
iv
ve
en
n a
ab
bo
ov
ve
e
i
i)
) F
Fo
or
r t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f f
fl
li
ip
pp
pi
in
ng
g a
a c
co
oi
in
n t
th
he
e p
po
os
ss
si
ib
bl
le
e o
ou
ut
tc
co
om
me
es
s a
ar
re
e e
ei
it
th
he
er
r H
He
ea
ad
d (
(H
H)
) o
or
r
T
Ta
ai
il
l (
(T
T)
).
. H
He
en
nc
ce
e,
, t
th
he
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e w
wi
il
ll
l b
be
e S
S =
={
{H
H,
, T
T }
}
i
ii
i)
) F
Fo
or
r t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f r
ro
ol
ll
li
in
ng
g a
a d
di
ie
e t
th
he
e p
po
os
ss
si
ib
bl
le
e o
ou
ut
tc
co
om
me
es
s a
ar
re
e t
th
he
e s
si
ix
x s
si
id
de
es
s o
of
f t
th
he
e d
di
ie
e
t
th
ha
at
t a
ar
re
e n
nu
um
mb
be
er
re
ed
d f
fr
ro
om
m 1
1 t
to
o 6
6.
. T
Th
hu
us
s,
, t
th
he
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e w
wi
il
ll
l b
be
e S
S=
={
{1
1,
,2
2,
,3
3,
,4
4,
,5
5,
,6
6}
}
i
ii
ii
i)
) T
To
os
ss
s a
a c
co
oi
in
n t
tw
wi
ic
ce
e a
an
nd
d o
ob
bs
se
er
rv
ve
e t
th
he
e f
fa
ac
ce
e u
up
p.
. T
Th
he
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e o
of
f t
th
hi
is
s e
ex
xp
pe
er
ri
im
me
en
nt
t i
is
s S
S
=
={
{H
HH
H,
, H
HT
T,
, T
TH
H,
, T
TT
T }
}.
. W
Wh
ha
at
t w
wo
ou
ul
ld
d b
be
e t
th
he
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e i
if
f y
yo
ou
u w
we
er
re
e t
to
o t
to
os
ss
s t
th
hr
re
ee
e
t
ti
im
me
es
s?
?
i
iv
v)
) F
Fo
or
r t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f d
dr
ra
aw
wi
in
ng
g a
a c
ca
ar
rd
d f
fr
ro
om
m a
a d
de
ec
ck
k o
of
f c
ca
ar
rd
ds
s t
th
he
e f
fo
ol
ll
lo
ow
wi
in
ng
g s
se
et
ts
s c
ca
an
n b
be
e
p
po
os
ss
si
ib
bl
le
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
es
s.
.
S
S=
={
{H
He
ea
ar
rt
t,
, S
Sp
pi
id
de
er
r,
, D
Di
ia
am
mo
on
nd
d,
, F
Fl
lo
ow
we
er
r}
}
S
S=
={
{1
1s
s,
, 2
2s
s,
, 3
3s
s,
, …
… ,
, Q
Qs
s,
, K
Ks
s,
, j
jo
ok
ke
er
rs
s}
}
S
S=
={
{B
Bl
la
ac
ck
k c
ca
ar
rd
d,
, R
Re
ed
d c
ca
ar
rd
d}
}
A
As
s t
th
he
e l
la
as
st
t c
ca
as
se
e o
of
f t
th
he
e a
ab
bo
ov
ve
e e
ex
xa
am
mp
pl
le
e i
in
nd
di
ic
ca
at
te
es
s i
it
t i
is
s p
po
os
ss
si
ib
bl
le
e t
to
o h
ha
av
ve
e m
mo
or
re
e t
th
ha
an
n o
on
ne
e s
sa
am
mp
pl
le
e s
sp
pa
ac
ce
e
f
fo
or
r a
a g
gi
iv
ve
en
n r
ra
an
nd
do
om
m e
ex
xp
pe
er
ri
im
me
en
nt
t.
. H
He
en
nc
ce
e,
, w
we
e c
ca
an
n s
st
ta
at
te
e t
th
he
e f
fo
ol
ll
lo
ow
wi
in
ng
g r
re
em
ma
ar
rk
k.
.
There are different types of sample space:-
9 Finite sample space: if they have a finite number of elements. Example:
• Tossing a coin. S={Heads, Tails}
• Throwing a die. S={1, 2, 3, 4, 5, 6}
• Throwing a coin and a die. S={H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
9 Countably infinite sample space: if there is a one-to-one mapping between the elements
of S and the natural numbers;
• S={0, 1, 2, 3, …}
• S={ … -3, -2, -1, 0, 1, 2, 3, …}

7. Page 7 of 18
• S={p: p is prime} = {2, 3, 5, 7, 11, 13, 17, …}
• S={1/r: r is a positive natural number} = {1/2, 1/3, 1/4, 1/5, … }
9 Uncountable sample space: if there are an infinite number of elements in S and there is
not a one-to-one mapping between the elements and the natural numbers.
• S=(0, ∞)
• S=(–∞, ∞)
• S={(x, y): 0≤ x≤ 1; y>0}
2. Outcome: The result of a single trial of experiment
3. Event: It is a subset of sample space. It is a statement about one or more outcomes of a
random experiment. It is denoted by capital letter.
E
Ex
xa
am
mp
pl
le
es
s:
:
¾
¾ I
In
n t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f f
fl
li
ip
pp
pi
in
ng
g a
a d
di
ie
e w
we
e c
ca
an
n d
de
ef
fi
in
ne
e a
an
n e
ev
ve
en
nt
t E
E1
1=
=H
He
ea
ad
ds
s s
sh
ho
ow
ws
s u
up
p=
={
{H
H}
}
¾
¾ I
In
n t
th
he
e e
ex
xp
pe
er
ri
im
me
en
nt
t o
of
f r
ro
ol
ll
li
in
ng
g a
a d
di
ie
e w
we
e c
ca
an
n d
de
ef
fi
in
ne
e a
an
n e
ev
ve
en
nt
t C
C2
2=
=T
Th
he
e n
nu
um
mb
be
er
r i
is
s o
od
dd
d {
{1
1,
,3
3,
,5
5}
}
¾
¾ I
In
n t
to
os
ss
si
in
ng
g a
a c
co
oi
in
n t
tw
wi
ic
ce
e w
we
e c
ca
an
n d
de
ef
fi
in
ne
e a
an
n e
ev
ve
en
nt
t D
D1
1=
=A
At
t l
le
ea
as
st
t o
on
ne
e h
he
ea
ad
d o
oc
cc
cu
ur
rs
s=
={
{H
HH
H,
, T
TT
T,
,
T
TH
H}
}
N
No
ot
te
e:
:
9 If S has n members, then there are exactly 2n
subsets or events.
9 Recall that any set is the subset of itself. Accordingly, the sample space S by itself is an
event called a sure event or certain event.
9 An event occurs if any one of its elements turns out to be the outcome of the experiment.
9 Empty set (φ) is called an impossible event.
9 Any event which consists of a single outcome in the sample space is called an
elementary or simple event whereas events which consist of more than one outcome are
called compound events.
4
4.
. M
Mu
ut
tu
ua
al
ll
ly
y E
Ex
xc
cl
lu
us
si
iv
ve
e e
ev
ve
en
nt
ts
s:
: T
Tw
wo
o e
ev
ve
en
nt
ts
s A
A a
an
nd
d B
B a
ar
re
e s
sa
ai
id
d t
to
o b
be
e m
mu
ut
tu
ua
al
ll
ly
y e
ex
xc
cl
lu
us
si
iv
ve
e i
if
f t
th
he
ey
y
c
ca
an
nn
no
ot
t o
oc
cc
cu
ur
r s
si
im
mu
ul
lt
ta
an
ne
eo
ou
us
sl
ly
y (
(t
to
og
ge
et
th
he
er
r)
) o
or
r A
A ∩
∩ B
B =
= φ
φ
E
Ex
xa
am
mp
pl
le
e:
: E
Ex
xp
pe
er
ri
im
me
en
nt
t -
-T
To
os
ss
s a
a c
co
oi
in
n t
tw
wi
ic
ce
e
S
S=
= {
{H
HH
H,
, H
HT
T,
, T
TH
H,
, T
TT
T}
}
L
Le
et
t A
A=
= T
Tw
wo
o h
he
ea
ad
ds
s o
oc
cc
cu
ur
r {
{H
HH
H}
}

8. Page 8 of 18
B
B=
= T
Tw
wo
o t
ta
ai
il
ls
s o
oc
cc
cu
ur
r {
{T
TT
T}
}
C
C=
= A
At
t l
le
ea
as
st
t o
on
ne
e h
he
ea
ad
d o
oc
cc
cu
ur
r {
{H
HH
H,
, H
HT
T,
, T
TH
H}
}
A
A ∩
∩ B
B=
= φ
φ ⇒
⇒ A
A a
an
nd
d B
B a
ar
re
e m
mu
ut
tu
ua
al
ll
ly
y e
ex
xc
cl
lu
us
si
iv
ve
e e
ev
ve
en
nt
ts
s
B
B ∩
∩ C
C=
= φ
φ ⇒
⇒ B
B a
an
nd
d C
C a
ar
re
e m
mu
ut
tu
ua
al
ll
ly
y e
ex
xc
cl
lu
us
si
iv
ve
e e
ev
ve
en
nt
ts
s
A
A ∩
∩ C
C=
= {
{H
HH
H}
} ⇒
⇒ A
A a
an
nd
d C
C a
ar
re
e n
no
ot
t m
mu
ut
tu
ua
al
ll
ly
y e
ex
xc
cl
lu
us
si
iv
ve
e
5. Independent Events: Two events are said to be independent if the occurrence of one does
not affect probability of the other occurring.
6. Dependent Events: Two events are dependent if the first event affects the outcome or
occurrence of the second event in a way the probability is changed.
7
7.
. C
Co
om
mb
bi
in
na
at
ti
io
on
n o
of
f e
ev
ve
en
nt
ts
s
a) Union: The event A∪ B occurs if either A or B or both occur.
I
If
f A
Ai
i,
, i
i=
=1
1,
, 2
2,
, 3
3…
…,
, n
n i
is
s a
an
ny
y f
fi
in
ni
it
te
e c
co
ol
ll
le
ec
ct
ti
io
on
n o
of
f e
ev
ve
en
nt
ts
s,
, t
th
he
en
n U
n
i
i
A
1
=
o
oc
cc
cu
ur
rs
s i
if
f a
at
t l
le
ea
as
st
t o
on
ne
e o
of
f
t
th
he
e e
ev
ve
en
nt
ts
s (
(A
Ai
i)
) o
oc
cc
cu
ur
rs
s.
.
b) Intersection: The event A ∩ B occurs if both A and B occur.
I
If
f A
Ai
i,
, i
i=
=1
1,
,2
2,
,3
3,
,…
…,
,n
n i
is
s a
an
ny
y f
fi
in
ni
it
te
e c
co
ol
ll
le
ec
ct
ti
io
on
n o
of
f e
ev
ve
en
nt
ts
s,
, t
th
he
en
n I
n
i
i
A
1
=
o
oc
cc
cu
ur
rs
s i
if
f a
al
ll
l A
Ai
i o
oc
cc
cu
ur
r.
.
c
c)
) Complement of an Event: the complement of an event A means nonoccurrence of A and
is denoted by A or A or A contains those points of the sample space which don’t belong
to A. S
S
)
) A
A∪
A
A/
/
=
=
S
S
)
) A
A∩
A
A/
/
=
=
φ
E
Ex
xe
er
rc
ci
is
se
e 1
1:
: A
A p
pe
er
rs
so
on
n i
is
s s
se
el
le
ec
ct
te
ed
d a
at
t r
ra
an
nd
do
om
m f
fr
ro
om
m a
a p
po
op
pu
ul
la
at
ti
io
on
n o
of
f a
a g
gi
iv
ve
en
n t
to
ow
wn
n
A
A:
: b
be
e t
th
he
e e
ev
ve
en
nt
t t
th
ha
at
t t
th
he
e p
pe
er
rs
so
on
n i
is
s m
ma
al
le
e
B
B:
: b
be
e t
th
he
e e
ev
ve
en
nt
t t
th
ha
at
t t
th
he
e p
pe
er
rs
so
on
n i
is
s u
un
nd
de
er
r 3
30
0 y
ye
ea
ar
rs
s
C
C:
: b
be
e t
th
he
e e
ev
ve
en
nt
t t
th
ha
at
t t
th
he
e p
pe
er
rs
so
on
n s
sp
pe
ea
ak
ks
s f
fo
or
re
ei
ig
gn
n l
la
an
ng
gu
ua
ag
ge
e
D
De
es
sc
cr
ri
ib
be
e t
th
he
e f
fo
ol
ll
lo
ow
wi
in
ng
g e
ev
ve
en
nt
ts
s s
sy
ym
mb
bo
ol
li
ic
ca
al
ll
ly
y
a
a)
) A
A m
ma
al
le
e u
un
nd
de
er
r 3
30
0 y
ye
ea
ar
rs
s w
wh
ho
o d
do
oe
es
s n
no
ot
t s
sp
pe
ea
ak
k f
fo
or
re
ei
ig
gn
n l
la
an
ng
gu
ua
ag
ge
e.
. (
(A
A ∩
∩ B
B)
) ∩
∩ C
C/
/

9. Page 9 of 18
b
b)
) A
A f
fe
em
ma
al
le
e w
wh
ho
o i
is
s e
ei
it
th
he
er
r u
un
nd
de
er
r 3
30
0 o
or
r s
sp
pe
ea
ak
ks
s f
fo
or
re
ei
ig
gn
n l
la
an
ng
gu
ua
ag
ge
e.
. A
A/
/
∩
∩ (
(B
B ∪
∪ C
C)
)
c
c)
) A
A p
pe
er
rs
so
on
n w
wh
ho
o i
is
s e
ei
it
th
he
er
r u
un
nd
de
er
r 3
30
0 o
or
r f
fe
em
ma
al
le
e b
bu
ut
t n
no
ot
t b
bo
ot
th
h.
. (
(B
B ∪
∪ A
A/
/
)
) ∩
∩ (
(B
B ∩
∩ A
A/
))c
d
d)
) M
Ma
al
le
e w
wh
ho
o i
is
s e
ei
it
th
he
er
r u
un
nd
de
er
r 3
30
0 o
or
r s
sp
pe
ea
ak
ks
s f
fo
or
re
ei
ig
gn
n l
la
an
ng
gu
ua
ag
ge
e b
bu
ut
t n
no
ot
t b
bo
ot
th
h.
.
A
A ∩
∩ [
[(
(B
B ∪
∪ C
C)
) ∩
∩ (
(B
B ∩
∩ C
C)
)/
/
]
]
E
Ex
xe
er
rc
ci
is
se
e 2
2:
: L
Le
et
t A
A,
, B
B a
an
nd
d C
C b
be
e t
th
hr
re
ee
e e
ev
ve
en
nt
ts
s a
as
ss
so
oc
ci
ia
at
te
ed
d w
wi
it
th
h a
an
n e
ex
xp
pe
er
ri
im
me
en
nt
t.
. D
De
es
sc
cr
ri
ib
be
e t
th
he
e
f
fo
ol
ll
lo
ow
wi
in
ng
g i
in
n s
sy
ym
mb
bo
ol
ls
s.
.
a
a)
) a
at
t l
le
ea
as
st
t o
on
ne
e o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
rs
s
b
b)
) o
on
nl
ly
y B
B o
oc
cc
cu
ur
rs
s
c
c)
) e
ex
xa
ac
ct
tl
ly
y o
on
ne
e o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
r
d
d)
) e
ex
xa
ac
ct
tl
ly
y t
tw
wo
o o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
r
e
e)
) a
at
t l
le
ea
as
st
t t
tw
wo
o o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
r
f
f)
) a
at
t m
mo
os
st
t t
tw
wo
o o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
r
g
g)
) n
no
on
ne
e o
of
f t
th
he
e e
ev
ve
en
nt
ts
s o
oc
cc
cu
ur
r
A
An
ns
sw
we
er
r:
:
a
a)
) A
AU
UB
BU
UC
C b
b)
) C
A
B ′
∩
′
∩ c
c)
) )
(
)
(
)
( A
B
C
C
A
B
C
B
A ′
∩
′
∩
∪
′
∩
′
∩
∪
′
∩
′
∩
d
d)
) )
(
)
(
)
( A
B
C
C
A
B
C
B
A ∩
′
∩
∪
∩
′
∩
∪
′
∩
∩
e
e)
) ))
)
(
)
)
(
)
)
(( A
B
C
B
C
A
C
B
A ∪
∩
∪
∪
∩
∪
∪
∩
f
f)
) C
B
A
C
B
A ′
∪
′
∪
′
=
′
∩
∩ )
( g
g)
) )
( C
B
A ′
∩
′
∩
′
1.4. Counting Techniques
In order to calculate probabilities, we have to know
• The number of elements of an event
• The number of elements of the sample space.
That is, in order to judge what is probable
, we have to know what is possible.
• In order to determine the number of outcomes, one can use several rules of counting.
) The addition rule
) The multiplication rule
) Permutation rule
) Combination rule
I. Addition Rule
Let E1 and E2 be any two mutually exclusive events (i.e. there are no common outcomes).
(A ∩ B∩C)
∪

10. Page 10 of 18
Let an event E describes the situation where either event E1 or event E2 will occur. Then
the number of times event E will occur can be given by the expression:
n(E) = n(E1) + n(E2)
where
n(E) = Number of outcomes of event E
n(E1) = Number of outcomes of event E1
n(E2) = Number of outcomes of event E2
In general, if there are k procedures and the procedure can be performed in ways,
1, 2, … . , , then the number of ways in which we may perform procedure 1 or 2 or….. or
procedure k is given by . ∑ , assuming no procedures may be
performed together.
Example 1: Suppose that we are planning a trip and deciding between bus and train
transportation. If there are three bus routes and two train routes. How many different routes are
available for the trip?
Solution: Let A= {an event for bus transportation}, B= {an event for train transportation}.
Then, we have 3 and 2; assuming that we are going to use only one route, by the
addition rule we have 3 2 5 different possible routes are available for the trip.
Example 2: Consider a set of numbers S = {-4, -2, 1, 3, 5, 6, 7, 8, 9, 10}
Let the events E1, E2 and E3 be defined as:
E = choosing a negative or an odd number from S;
E1= choosing a negative number from S;
E2 = choosing an odd number from S.
Find n(E).
Solution:
E1 and E2 are mutually exclusive events (i.e. no common outcomes).
n(E) = n(E1) + n(E2)
= 2 + 5
= 7
Example 3
In how many ways can a number be chosen from 1 to 22 such that it is a multiple of 3 or 8?
Solution:

11. Page 11 of 18
Here, E1 = multiples of 3:
E1 = {3, 6, 9,12, 15, 18, 21}
n(E1) = 7
E2 = multiples of 8:
E2 = {8, 16}
n(E2) = 2
Events E1 and E2 are mutually exclusive.
n(E) = n(E1) + n(E2) = 7 + 2 = 9
II. The Multiplication Rule
If there are k procedures and the procedure can be performed in ways, 1, 2, … . , , then
the procedure consisting of procedure 1 followed by procedure 2,…, followed by procedure k
may be performed in … ∏ ways.
Example 1: Suppose that there are three different types of meal for lunch and four different
types of soft drinks. How many choices of meal and soft drinks can be made?
Solution: 3 4 hence 3 4 12 different choices of meal and soft
drinks can be made.
Example 2: How many two digit numerals can be written by choosing the ten’s digit from
A={1,3,5,7,9} and the units digit from B= {2,4,6,8}
Solution: The selection consists of two steps where the 1 st
can be made in 5 different ways for
the ten’s digits and for each of these the 2nd
can be made in 4 different ways for the units digit,
Hence the whole selection one after the other can be made in 5 x 4 different ways. i.e. there are
20 two digit numerals
i.e. = 5 and = 4
Example 3: The digits 0, 1, 2, 3, and 4 are to be used in 4 digit identification card. How many
different cards are possible if
a) Repetitions are permitted.
b) Repetitions are not permitted.
Solution:
a) There are four steps
1st
digit 2nd
digit 3rd
digit 4th
digit
5 5 5 5

12. Page 12 of 18
1. Selecting the 1st
digit, this can be made in 5 ways.
2. Selecting the 2nd
digit, this can be made in 5 ways.
3. Selecting the 3rd
digit, this can be made in 5 ways.
4. Selecting the 4th
digit, this can be made in 5 ways.
Hence, 5 5 5 5 625 different cards are possible.
b) There are four steps
1st
digit 2nd
digit 3rd
digit 4th
digit
5 4 3 2
1. Selecting the 1st
digit, this can be made in 5 ways.
2. Selecting the 2nd
digit, this can be made in 4 ways.
3. Selecting the 3rd
digit, this can be made in 3 ways.
4. Selecting the 4th
digit, this can be made in 2 ways.
Hence, 5 4 3 2 120 different cards are possible.
Example 4: There are 6 roads between A and B and 4 roads between B and C.
a. In how many ways can one drive from A to C by way of B?
b. In how many ways can one drive from A to C and back to A, passing through B on both
trip?
c. In how many ways can one drive the circular trip described in (b) without using the same
road more than once?
Ans. 24, 576 and 360 resp
III. Permutation
An arrangement of n objects in a specified order is called permutation of the objects.
Permutation Rules:
1. The number of permutations of distinct objects taken all together is n!
Where ! 1 2 … 2 1. Note that: 0! 1
2. The arrangement of objects in a specified order using objects at a time is called the
permutation of objects taken r objects at a time. It is written as n Pr and the formula is
( )!
r
n
n!
=
Pr
n
−
3. The number of distinct permutation of n objects in which are alike, are alike ---- etc is
!
*
....
!*
!* 2
1 k n
n
n
n! ,where n = n1+n2+...+nk

13. Page 13 of 18
4. An arrangement of distinct objects around a circle is (n-1)! ways.
Example 1: Suppose we have the letters A, B, C, and D
a) How many permutations are there by taking all the four?
b) How many permutations can be formed by taking two letters at a time?
Solution: a) Here 4, there are four distinct objects. Hence, there are 4! 24 permutations.
b) Here 4 2. There are
( )
12
2
24
2
4
4
2
4 =
=
− !
!
=
P permutations.
Example 2: How many different permutations can be made from the letters in the word
“CORRECTION”?
Ans: 453600
Example 3: In how many ways can a party of 7 people arrange themselves
a) on a row of 7 chairs?
b) around a circular table?
c) How many ways are there if two of the persons are not allowed to sit next to each
other?
Ans.: 5040, 720 and 3600 respectively.
IV. Combination
A selection of r objects from n objects without regard to order is called combination.
Example: Given the letters A, B, C, and D list the permutation and combination for selecting
two letters.
Solution:
Permutation Combination
AB, BA, CA , DA,
AC, BC, CB, DB,
AD, BD, CD, DC
AB, BC,
AC, BD,
AD, DC
Note that: in permutation AB is different from BA. But in combination AB is the same as BA.
Combination Rule
The number of combinations of r objects selected from n objects is denoted by nСr or and is
given by the formula:
!
! !
(The arrangement is in a row)

14. Page 14 of 18
Example 1: In how many ways can a committee of 5 people be chosen out of 9 people?
Ans: 126 ways
Example 2: Among 15 clocks there are two defectives. In how m any ways can an inspector
choose three of the clocks for inspection so that:
a) there is no restriction,
b) none of the defective clocks is included,
c) only one of the defective clocks is included,
d) two of the defective clocks is included.
Ans: 455, 286, 156, and 13 ways, respectively.
Example 3: A delegation of four peopleis selected each year from a college to attend a meeting.
a. In haw many ways can the delegation be chosen if there are 12 eligible students?
b. In how many ways if two of the eligible students will not attend the meeting together?
c. In how many ways if two of the eligible students are married and will only attend the
meeting together?
12C4=495, 10C4=210 + 2C1*10C3=240=450, 10C4=210+ 2C2*10C0 = 211
Some remarks
1. When we select objects from distinct objects, we have objects unselected and
hence there are as many ways of selecting from as there are not selected objects.
Thus
a.
b.
2. Binomial Coefficients
T
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(a
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15. Page 15 of 18
S
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1.5.Definitions of probability
In any random experiment there is always uncertainty as to whether a particular event will or will
not occur. Thus, if is an event associated with an experiment, we cannot state with certainty
that an event will occur or will not occur. Hence, it becomes very important to try to associate
a number with the event A which will measure how likely is the event will occur in the defined
sample space. This leads to the theory of probability in which we are going to consider the
quantitative measure of uncertainty.
Approaches to Measuring Probability
There are four different conceptual approaches to study probability theory. These are:
9 The classical approach.
9 The frequentist approach.
9 The subjective approach.
9 The axiomatic approach.
A. The classical approach
This approach is used when:
) All outcomes are equally likely and mutually exclusive.
) Total number of outcomes is finite, say N.
Definition: If a random experiment with N equally likely outcomes is conducted and out of these
NA outcomes are favorable to the event A, then the probability that event A occurs denoted P (A)
is defined as:
o . of outcomes favorable to the event A
Total nuber outcomes
Example 1: A fair die is tossed once. What is the probability of getting
a) Number 4?
b) An odd number?

16. Page 16 of 18
c) Number greater than 4?
d) Either 1 or 2 or …. or 6
Example 2: A box of 80 candles consists of 30 defective and 50 non-defective candles. If 10
of these candles are selected at random, what is the probability?
a) All will be defective.
b) 6 will be non defective
c) All will be non defective
Solution:
Let A be the event that all will be defective.
The total ways in which will occur=
Hence,
0.00001825
Ans: 0.265, and 0.00624 for b and c, respectively.
Limitation: The classical approach cannot be employed if:
9 it is not possible to enumerate all the possible outcomes for an experiment.
9 the sample points (outcomes) are not mutually exclusive.
9 each and every outcomes is not equally likely.
B. The Frequentist Approach
This is based on the relative frequencies of outcomes belonging to an event.
Definition: The probability of an event A is the proportion of outcomes favorable to A in the
long run when the experiment is repeated under same condition.
lim
Example 1: If records show that 60 out of 100,000 bulbs produced are defective. What is the
probability of a newly produced bulb to be defective?
lim
60
100,000
0.0006
Limitations of frequentist approach:
9 If repeated trials are not possible we cannot use this approach
9 The limit as N approaches to infinity has no clear meaning (How large is large??)

17. Page 17 of 18
9 Generating infinite trials has cost imposition.
C. Subjective Approach
It is always based on some prior body of knowledge. Hence subjective measures of uncertainty
are always conditional on this prior knowledge.
Definition: Subjective probability is the degree of believe assigned to the occurrence of an event
by a particular individual.
Examples: - A medical Dr. may assign the probability that a patient recovers from a disease.
- Estimating the likelyhood that you will earn an A grade in this course.
D. Axiomatic Approach:
Let E be a random experiment and S be a sample space associated with E. With each event A
defined on S. We can associate a real number called the probability of A satisfying the following
properties called axioms of probability or postulates of probability.
1. 0 (i.e. the probability of any event is non-negative real number)
2. 1 (Sure event)
3. If and are two mutually exclusive events, the probability that one or the other occur
equals the sum of the two probabilities. i. e. .
4. If , , …. form a sequence of pair wise mutually exclusive events (meaning they
satisfy for all then
, … .
1.6.Derived theorems of probability
1. 0 for any sample space S
2. If ′ is the complement of event , then ′
1
3. If are any two events in S, then
4. If then
5. For any two events say, A and B, the probability that exactly one of the events A or B
occurs but not both is :
2

18. Page 18 of 18
E
Ex
xa
am
mp
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s
1
1)
) F
Fo
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s A
A a
an
nd
d B
B s
sh
ho
ow
w t
th
ha
at
t P
P(
(A
A ∪
∪ B
B)
) ≤
≤ P
P(
(A
A)
) +
+ P
P(
(B
B)
)
2
2)
) S
Su
up
pp
po
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se
e A
A a
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B a
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ev
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ts
s f
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or
r w
wh
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ch
h
P
P (
(A
A)
) =
=x
x P
P(
(B
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= y
y a
an
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d P
P(
(A
A ∩
∩ B
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=z
z
F
Fi
in
nd
d i
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. P
P (
(A
A/
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∪
∪ B
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ii
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P (
(A
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3
3)
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If
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= ¾
¾ a
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=1
1/
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4, caan
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A a
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d B
B b
be
e mutually exclusive?
4
4)
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If
f P
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Ac
=
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P(
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) =
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b S
Sh
ho
ow
w t
th
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at
t P
P(
(A
A ∩
∩ B
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) ≥
≥ 1
1-
-a
a-
-b
b
)