Math 3338: Probability (Fall 2006), September 4- September 8, 2006
Math 3338: Probability (Fall 2006)
Jiwen He
Section Number: 10853
http://math.uh.edu/˜jiwenhe/math3338fall06.html
Probability – p.1/9
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Conditional Probability & Multiplication Rule in Probability Theory by Jiwen He (UH, 2006), Exams of Probability and Statistics
A part of the lecture notes for the university of houston math 3338: probability course taught by jiwen he in fall 2006. It covers the concepts of conditional probability and the multiplication rule. Definitions, examples, and theorems to help students understand these concepts.
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Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
Math 3338: Probability (Fall 2006)
Jiwen He
Section Number: 10853
http://math.uh.edu/
˜jiwenhe/math3338fall06.html
Probability – p.1/
Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
2.4 Conditional Probability
Probability – p.2/
Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
Examples •^
2.24:
Let
A^
=^
{a line
A^
component is selected
},^ B
{the chosen component is defective
Line
B^
′B
P^ (
A) =
|A
|= |Ω|^
(^8 )
.^44
A^
′ A
P^ (
A|
B) =
P (A
∩B
) P^ ( B)
=^
(^218318)
=^
23
•^
2.25:
Let
A^
=^
{memory card purchased
}^ and
B
{battery purchased
}. Then
P^ (
A) =
.^6
,^
P^ (
B) =
.^4
,^
P^ (
A^ ∩
B
.^3
The conditional probabilities are
P^ (
A|
B) =
P
(A
∩^
B)
P^ (
B)
=^
.^3.^4
=^
.^75
,^
P^ (
B|
A) =
P
(A
∩^
B)
P^ (
A)
=^
.^3.^6
=^
.^5.
•^
2.26:
Reading habits with respect to “Art” (
A), “Book” (
B), and “Cinema” (
C):
P^ (
A|
B) =
P^ (A ∩B
) P^ (B
)^
=^
.^08.^23
.^348
P^ (
A|
B^
∪^ C
P (A
∩(B
∪C
))
P^ ( B∪
C)
=^
.04+
.05+
.^03 .^47
=^
.^255
P^ (
A|
A^ ∪
B^
∪^ C
P^ (A ∩(A
∪B
∪C
))
P^ ( A∪
B∪
C)
=^
P^ (A
)) P^ ( A∪
B∪
C)
.^14.^49
.^286
P^ (
A^ ∪
B|
C) =
P ((A
∪B
)∩C
)
P^ ( C)^
=^
.04+
.05+
.^08 .^37
=^
.^459 Probability – p.4/
Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
Multiplication Rule for
P
(A
B
•^
Multiplication rule:
P^ (
A^ ∩
B
)^
=^
P^ (
A|
B)
·^ P
(B
=^
P^ (
B|
A)
·^ P
(A
•^
2.27:
Four individuals are selected in random order for typing. Only type O+ is desired and only one of the four actually has this type. What is the probability that at least three individualsmust be typed to obtain the desired type? Let
{four blood types, in which only one type is O+
B^
=^
{first type is not O+
A^
=^
{second type is not O+
As three of the four types are not O+, and given that the first type is not O+, two of the three lefttypes are not O+, we have
P^ (
B) =
3 ,^4
P^ (
A|
B) =
The intersection of
A
and
B^
gives the event that at least three individuals are typed. The
multiplication rule now gives
P^ (
A^ ∩
B) =
P^
(A
|B
)^ ·^
P^ (
B) =
=^
.^5
Probability – p.5/
Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
The Law of Total Probability •^
Mutually exclusive:
Events
A
,^... 1
,^ A
arek
mutually exclusive
if no two have any common
outcomes, so that
Ai
∩^
Aj
∅^
for any
i, j
,... , k
•^
Exhaustive:
Events
A^1
,^...
,^ A
arek
exhaustive
if one
A
must occur, so thati
A^1
Ak
•^
Theorem:
Let
A^1
,^...
,^ A
be mutually exclusive and exhaustive events, so thatk
A^1
+^
Ak
. Then for any other event
B
P^ (
B) =
P^
(B
|A
)P 1
(A
P^ (
B|
Ak
)P
(A
) =k
kX i=
P^ (
B|
Ai
)P
(A
)i
•^
Proof:
From the partition of
B
B^
=^
B^
∩^ Ω =
B
∩^
(A
Ak
B
∩^
A^1
B^
∩^ A
=k
kX i=
B^
∩^ A
i
it follows that
P^ (
B) =
P^
kX( i=
B^
∩^ A
) =i
kX i=
P^ (
B^
∩^ A
) =i
kX i=
P^ (
B|
Ai
)^ ·^
P^ (
Ai
).^ Probability – p.7/
Jiwen He, University of Houston, [email protected] 3338: Probability (Fall 2006), September 4- September 8, 2006
Bayes’ Theorem •^
Theorem:
Let
A
,^1
,^ A
be mutually exclusive and exhaustivek
events with
P
(A
)^ i
for
i^
,... , k
. Then for any other event
B
for which
P
(B
)^ >
, we have, for
j^
,... , k
P^
(A
|Bi
P
(A
∩i
B
P^
(B
)^
P^
(B
|A
)^ i
·^ P
(A
)i
k j=
P
(B
|A
)j
·^ P
(A
)j
•^
Proof:
combine the total probability law and the multiplication
rule.
•^
Prior and posterior Probabilities:
The computation, provided by
Bayes’ theorem, of a posterior probability
P
(A
|Bi
)^ from given
prior probabilities
P
(A
)^ i
and conditional probabilities
P
(B
|A
),i
which occupies a central position in elementary probability.
Probability – p.8/