1
Lecture 2
ECE 514
¾
Using the
coin tossing experiment,
and from
experience we know that if we keep tossing a coin,
eventually, a head must show up, i.e.,
But
and using axiom 4
(generalization of axiom 3) of
probability
.
1
)
(
=
A
P
∪
∞
=
=
1
,
n
n
A
A
).
(
)
(
1
1
∑
∞
=
∞
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
n
n
n
n
A
P
A
P
A
P
∪
¾
In a fair coin since only one in 2
n
outcomes is in favor of
A
n
, we have
Agreeing
with
axiom 2 of probability
¾
In summary, we have a nonempty set
Ω
of outcomes (
elementary events), a
σ
field
F
of subsets of
Ω
(combining
to form events), and a probability measure
P
on the sets in
F
subject to the four axioms forming a triplet
(
Ω
,
F, P
)
i.e.
a
probability model
.
¾
The probability of more complicated events must follow
from this framework by deduction.
,
1
2
1
)
(
and
2
1
)
(
1
1
=
=
=
∑
∑
∞
=
∞
=
n
n
n
n
n
n
A
P
A
P
Conditional Probability and Independence
¾
In
N
independent trials, suppose
N
A
,
N
B
,
N
AB
denote the
number of times events
A
,
B
and
AB
occur respectively.
¾
Using
the frequency interpretation of probability, for
large
N
¾
In
N
A
occurrences of
A,
only
N
AB
of them are also found
in the
N
B
occurrences of
B,
giving
the ratio
.
)
(
,
)
(
,
)
(
N
N
AB
P
N
N
B
P
N
N
A
P
AB
B
A
≈
≈
≈
)
(
)
(
/
/
B
P
AB
P
N
N
N
N
N
N
B
AB
B
AB
=
=
as a measure of “the event
A
given that
B
has already
occurred”. We denote this conditional probability by
P
(
A

B
) =
Probability of “the event
A
given
that
B
has occurred”.
¾
We define
provided
¾
For a conditional probability to be a probability, it has to
satisfy all probability axioms discussed earlier.
,
)
(
)
(
)

(
B
P
AB
P
B
A
P
=
.
0
)
(
≠
B
P
We have
(i)
(ii)
since
Ω
B
=
B
.
(iii)
Suppose
Then
But
hence
satisfying all probability axioms , making a
cond.
Prob.
a legitimate probability measure.
,
0
0
)
(
0
)
(
)

(
≥
>
≥
=
B
P
AB
P
B
A
P
.
)
(
)
(
)
(
)
)
((
)

(
B
P
CB
AB
P
B
P
B
C
A
P
B
C
A
P
∪
=
∩
∪
=
∪
,
1
)
(
)
(
)
(
)
(
)

(
=
=
Ω
=
Ω
B
P
B
P
B
P
B
P
B
P
),

(
)

(
)
(
)
(
)
(
)
(
)

(
B
C
P
B
A
P
B
P
CB
P
B
P
AB
P
B
C
A
P
+
=
+
=
∪
.
0
=
∩
C
A
AB
BC
,
∩=
φ
).
(
)
(
)
(
CB
P
AB
P
CB
AB
P
+
=
∪
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Properties of Conditional Probability:
a. If
and
since if
then occurrence of
B
implies automatic
occurrence of the event
A
.
As an example, but
in a dice tossing experiment. Then
and
b. If
and
,
,
B
AB
A
B
=
⊂
1
)
(
)
(
)
(
)
(
)

(
=
=
=
B
P
B
P
B
P
AB
P
B
A
P
,
A
B
⊂
).
(
)
(
)
(
)
(
)
(
)

(
A
P
B
P
A
P
B
P
AB
P
B
A
P
>
=
=
,
,
A
AB
B
A
=
⊂
,
A
B
⊂
.
1
)

(
=
B
A
P
{outcome is even},
={outcome is 2},
A
B
=
¾
In a dice experiment,
so that
¾
The statement that
B
has occurred (outcome is even)
makes “outcome is 2” more likely than without that
information
¾
A conditional probability may be used to simplify the
probability of a complicated event (in terms of “simpler”
related events)
¾
Let
be pair wise disjoint and their union is
Ω
. Thus
and
Thus
.
B
A
⊂
.
1
Ω
=
=
∪
n
i
i
A
n
A
A
A
,
,
,
2
1
"
,
φ
=
j
i
A
A
.
)
(
2
1
2
1
n
n
BA
BA
BA
A
A
A
B
B
∪
∪
∪
=
∪
∪
∪
=
"
"
{outcome is 2},
={outcome is even},
A
B
=
¾
But
so that the union
expression above yields
¾
Along with
the notion of conditional probability,
we
introduce the notion of “independence” of events
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 Fall '08
 Krim
 Probability, Probability theory

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