2
C
iapter
2
Fundamentals
of
Probabdity Models
2.3
Mathematics
of
Probability
53
For
three
events,
the
multiplication
rule
would
give,
pi
)
and
P(fr)
P(E
1
E
2
3
)
P(E
1
E
2
E
2
)P(E
E)
(2.14a)
7
7
=
P(E
1
IE
2
E
3
)P(E
2
)
At the
intersection,
if
a
vehicle
is
definitely
makine
a
turn, the
probability
that
it
will
be
a
right
turn
whereas
if
the
three
events
are
statistically independent,
is
(observe that
the
three alternatwe directions
are
mutually
exelusise>
P[(L
frl
P(L
fr
3
)
P(E
1
E
2
3
)
=
P(E
1
)P(E
2
3
)
(2.15a)
—
P(E
3
L)
P(L
L
3
)
We
would
expect
that
if
two
events
E
1
and
E
2
are
statistically
independent, their
comple
PU
2>
ments
E
1
and
E
2
would
also be
statistically independent;
i.e.,
P()
P(1
i
P(E
1
E
2
)
=
P(E
1
)P(E
2
)
(2.16)
On
the
other
hand,
if
the
sehicle
is
definitely
making
i
turn
at
the
intersection
the
probability that
it
In
fact
we
can verify
this
assertion
in
the
case
of
two
events
as
follows:
will
not
turn
right,
according
t
Lq
2
12
i
P(E
1
E
2
)
=
P(E
1
UE
2
)
=
I
—
P(E
U
L
2
)
P(J
L
f)
1
=
1
IP(Ei)
+
P(E)
—
P(E
1
)P(E
2
)j
=
[I
—
P(E
1
)j[l
—
P(E
2
)1
=
P(E
1
)P(E
2
)
Statistical Independence
We
might
emphasize
that
all
the
mathematical
rules
pertaining
to
the
probabilities
of
If
the
oceu
Ten
e
or
nonoe
urr
nrc,
f
n
a
n
does
ra
t
iffeet
the
probability
of
occurrence
events apply
equally
to
the
conditional
probabilities
of
events that
are
conditioned
on
the
ot
another
eent
the
w
serts
e
itt
co11
idipu
dent,
Ir
othLl
words
the
pr
)bability
same
reconstituted
sample
space,
including
the
addition
rule
and
the
multiplication
rule.
In
of
occurrence
of
ne
ent
do
no
per
I
rn
t
e
rr nec
or
nonoccurrence
of
another
particular,
we
may
observe
the
following:
event
I
he
for
if
tw(
escnts
I
and
£
e
a
i
tically
mndeoendent
P(E
1
U
E
2
A)
=
P(E
1
A)
+
P(E
2
A)
P(E
1
E
2
IA)
P(E
1
E
2
IA)
=
P(EilEaA)P(E
2
IA)
and
and
if
E
1
and
E
2
are
statistically
independent
events, given event
A,
P(L
1
E
2
IA)
=
P(E
1
jA)P(E2IA)
It
night
b
piude
i
to
po
nt
u
lkr
K
re
bit
cer
rv
that
ire
statistual
i
indepet
dent
ersi
s
those
tCat
e
nuttu
Its
It
cc
he
dill
erer
cc
is
profound
and
there
bility
f
heir
join
occ
i
en
e
v
creas
tw
a
ils
a
e
rnutu
2llv
cx
lusi
s
hen
tf
a
Joint
subjected
to
a
force
F
=
300
kg.
occurrence
s
i
np
)s
ihi
s
he
cc
K
1
e
a
ant
p
eeludes
+
at
n
e
of
the
o
hr
i.
In
2th
s
o
Us
(T
11
1
f
ifl(
F
2
are m
miii
lv
ii
six
nal
x
staiist
ml
Link
1
Link2
independence
of
two
w
more
‘vents
Pc
i
5
t(
the
probabili
y
of
the
j
nnt
esent
vh
‘e
is
F=300
kg
F300
kg
mutual
exclusie
ess
refers
the
del
mit
o
m
af
the
e
en
Figure
E2.19
A
twolink
chain.
p.3.3
The
Muitiphcation
Rue
If
the
fracture
strength
of
a
link
is
less
than
300
kg,
it
will fail
by
fracture.
Suppose
that
the
probability
of
this
happening
to
either
of
the
two
links
is
0.05.
Clearly. the
chain
will
fail
if
one
or both
of
the
From
iq
2
1
,
it
folk
w
tI
at
t
s
or
aab’lity
f
thej
i
t
e
mt
s
two
links
should
fail
by
fracture.
‘Jo
determine
the
probability
of
failure
of
the
chain,
define
U
PiE
PT
3
F
1

fracture
of
link
1
F
2
=
fracture
of
link
2
tipli orion
tule
be
‘om
P(E
1
U
E
2
)
=
P(E
1
)
+
P(E
2
)
P(E
1
E
2
)
P(E
1
E
7
)
=
PU’i)P(I
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 '11
 MTODOROVSKA
 Conditional Probability, Probability, Probability theory, PROBABILITY MODELS

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