chapter 1 probability spaces 主講人 : 虞台文. content sample spaces and events event...
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Chapter 1Probability Spaces
主講人 :虞台文
Content Sample Spaces and Events Event Operations Probability Spaces Conditional Probabilities Independence of Events Reliabilities Bayes’ Rule
Chapter 1Probability Spaces
Sample Spacesand
Events
Definitions Sample Spaces and Events
The set of all possible outcomes of a random
experiment is called the sample space,
denoted by , of that experiment.
An element is called a sample point.
A subset A is called an event.
Example 1
The set of all possible outcomes of a random experiment is called the sample space, denoted by , of that experiment.
An element is called a sample point.
A subset A is called an event.
The set of all possible outcomes of a random experiment is called the sample space, denoted by , of that experiment.
An element is called a sample point.
A subset A is called an event.
Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose face number is less than 4
2 4,5,6A an event whose face number is lager than 3
3 1,3,5A an event whose face number is odd
4 2,4,6A an event whose face number is even
Notations
A
a sample space
an event
an sample point
( )P A
( )P
the probability for the occurrence of event A
P
Example 2
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. Suppose that the die in Example 1 is fair. Find the probability of each event.
2. Suppose that the die is unfair, and with
Find the probability of each event.
14(1) (6)P P 1
16(3) (4)P P 316(2) (5)P P
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
Example 2
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. Suppose that the die in Example 1 is fair. Find the probability of each event.
2. Suppose that the die is unfair, and with
Find the probability of each event.
14(1) (6)P P 1
16(3) (4)P P 316(2) (5)P P
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
12
12
12
12
Example 2
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. Suppose that the die in Example 1 is fair. Find the probability of each event.
2. Suppose that the die is unfair, and with
Find the probability of each event.
14(1) (6)P P 1
16(3) (4)P P 316(2) (5)P P
1( )P A
2( )P A 3( )P A
4( )P A
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
12
12
12
12
Example 2
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. Suppose that the die in Example 1 is fair. Find the probability of each event.
2. Suppose that the die is unfair, and with
Find the probability of each event.
14(1) (6)P P 1
16(3) (4)P P 316(2) (5)P P
1( )P A
2( )P A 3( )P A
4( )P A
1( ) ?P A
2( ) ?P A 3( ) ?P A
4( ) ?P A
12
12
12
12
31 1 14 16 16 2
31 1 116 16 4 2
31 1 14 16 16 2 3 1 1 1
16 16 4 2
Example 3 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
sample space of a bulb life
the life of a bulb is not less than 10 months
the life of a bulb is not less than 10 but less than 20 months
the life of a bulb is less than 10 months
the life of a bulb is larger than 15 but not larger than 30 months
Example 3 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
sample space of a bulb life
the life of a bulb is not less than 10 months
the life of a bulb is not less than 10 but less than 20 months
the life of a bulb is less than 10 months
the life of a bulb is larger than 15 but not larger than 30 months
0 5 10 15 20 25 30 35 40 45 50
Example 3 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1( ) ?P B
2( ) ?P B
3( ) ?P B
4( ) ?P B
Example 4
Tossing three balanced coins
1. Write the sample space of this experiment.
2. Write the event A to denote that at least two coins land heads.
3. P(A) =?
Example 4
Tossing three balanced coins
1. Write the sample space of this experiment.
2. Write the event A to denote that at least two coins land heads.
3. P(A) =?
, , , ,
, , ,
HHH HHT HTH HTT
THH THT TTH TTT
, , ,A HHH HHT HTH THH
12( )P A
,HHH ,HHT ,HTH ,HTT
,THH ,THT ,TTH TTT
Example 4
Tossing three balanced coins
1. Write the sample space of this experiment.
2. Write the event A to denote that at least two coins land heads.
3. P(A) =?
, , , ,
, , ,
HHH HHT HTH HTT
THH THT TTH TTT
, , ,A HTT THT TTH TTT
12( )P A
,HHH ,HHT ,HTH ,HTT
,THH ,THT ,TTH TTT
The method to define a sample space is not unique, e.g.,
0 3 ,1 2 ,2 1 ,3 0H T H T H T H T
Chapter 1Probability Spaces
Events Operations
Event Operations
Intersection And
Union Or
Complement Not
Intersection () And
Venn Diagram
A B
AB
Example 5
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. A1 A3 = {1, 3}
2. A2 A4 = {4, 6}
3. A1 A2 =
The face number is less than 4 and odd.
The face number is larger than 3 and even.
A null event.
Example 5
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. A1 A3 = {1, 3}
2. A2 A4 = {4, 6}
3. A1 A2 =
The face number is less than 4 and odd.
The face number is larger than 3 and even.
A null event.
Remark: Two events A and B are
said to be mutually exclusive if A ∩
B = .
Example 5
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. A1 A3 = {1, 3}
2. A2 A4 = {4, 6}
3. A1 A2 =
1 3( ) ?P A A
2 4( ) ?P A A
1 2( ) ?P A A
13
13
0
Assume the die is fair.
Example 6 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1 3B B A null event.
Example 6 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
2 4 (15,20)B B The bulb life is lager than 15 but less than 20 months
Example 6 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1 2 [10,20)B B 2B
Union () Or
Venn Diagram
A B
AB
Example 7
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1. A1 A3 = {1, 2, 3, 5}
2. A2 A4 = {2, 4, 5, 6}
3. A1 A2 = 1
The face number is less than 4 or odd.
The face number is larger than 3 or even.
A universal event.
Example 8 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1 3 2B B A universal event.
Example 8 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
2 4 [10,30]B B The bulb life is not less than 10 but not large than 30 months
Example 8 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1 2 [10, )B B 1B
Complement Not
Venn Diagram
AAc
{ | , }cA A A A
Example 9-1
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
Example1. Tossing a die.
1 1,2,3,4,5,6 sample space of tossing a die
1 1,2,3A an event whose f ace number is less than 4
2 4,5,6A an event whose f ace number is lager than 3
3 1,3,5A an event whose f ace number is odd
4 2,4,6A an event whose f ace number is even
1 4,5,6cA 2A
The face number is not less than 4.
Example 9-2 Bulb Life (Months)
2 [0, )
1 [10, )B
2 [10,20)B
3 [0,10)B
4 (15,30]B
0 5 10 15 20 25 30 35 40 45 50
1 [0,10)cB 3B
Laws of Event Algebra
A B C A B C
A B C A B C
1. Associative laws
2. Commutative laws
3. Distributive laws
A B B A
A B B A
A B C A B A C
A B C A B A C
Laws of Event Algebra
A A
A A
4. Identity laws
5. Complementation laws
6. Idempotent laws
c
c
A A
A A
A A A
A A A
Laws of Event Algebra
A
A
7. Domination laws
8. Absorption laws
9. De Morgan’s laws
A A B A
A A B A
c c c
c c c
A B A B
A B A B
More on De Morgan’s laws
cc
i ii i
cc
i ii i
A A
A A
c c c c
c c c c
A B C A B C
A B C A B C
c c c
c c c
A B A B
A B A B
Chapter 1Probability Spaces
Probability Spaces
-Field
A nonempty collection of subsets A is called -
field of a set provided that the following two
properties hold:
1. A
2.
cA A A A
an, 1, d2, i i ii iA i A A A A A
Example 10
Let = {1, 2, 3, 4, 5, 6}.
1. Let A1 = {A0 = , A1 = {1, 2, 3}, A2 = {4, 5, 6}, A3 = } .
Whether or not A1 forms a -field of ?
2. Let A2 = {A1 = {1, 6}, A2 = {2, 5}, A3 = {3, 4}}.
Add minimum number of subsets of into A2 such that A2 becomes a -field of .
The Axioms of Probability
A probability measure P on a -field of subset A of a set is a real-valued function having domain A satisfying the following properties:
1. P() = 1;2. P(A) 0 for all A A;3. If Ai A, i=1,2,. . . are
mutually disjoint then ( )i iii
P A P A
AA1A2 A3
A4
Definition Probability Space
A probability measure P on a -field of subset A of a set is a real-valued function having domain A satisfying the following properties:
1. P() = 1;2. P(A) 0 for all A A;3. If Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
A probability space, denoted by (, A, P), is a set , a -field of subsets A, and a probability measure P defined on A.
Example 11
= {1, 2, 3, 4, 5, 6}.
A = {A0 = , A1 = {1, 2, 3}, A2 = {4, 5, 6}, A3 = }
P (A0) = 0, P (A1) = 1/2, P (A2) = 1/2, P (A3) = 1
(, A, P) A probability space?
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
Example 11
= {1, 2, 3, 4, 5, 6}.
A = {A0 = , A1 = {1, 2, 3}, A2 = {4, 5, 6}, A3 = }
P (A0) = 0, P (A1) = 1/3, P (A2) = 2/3, P (A3) = 1
(, A, P) A probability space?
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
Example 11
= {1, 2, 3, 4, 5, 6}.
A = {A0 = , A1 = {1, 2, 3}, A2 = {4, 5, 6}, A3 = }
P (A0) = 1/3, P (A1) = 1/3, P (A2) = 1/3, P (A3) = 1
(, A, P) A probability space?
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
Example 11
= {1, 2, 3, 4, 5, 6}.
A = {A0 = , A1 = {1, 2, 3}, A2 = {4, 5, 6}, A3 = }
P (A0) = 0, P (A1) = 1/3, P (A2) = 1/3, P (A3) = 2/3
(, A, P) A probability space?
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
Theorem 1-1
For any event A, P(Ac) = 1 P(A).
Pf) , c cA A A A Facts:
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
1. P() = 1;2. P(A) 0 f or all A A;3. I f Ai A, i=1,2,. . . are
mutually disjoint then ( )i iiiP A P A
( ) ( )c cP A A P A P A ( )P 1
( ) 1 ( )cP A P A
Theorem 1-2
For any two events A and B,
P(A B) = P(A) + P(B) P(A B)
Pf) ( )cA B A A B
A BcA BA B
( ) ( )cB A B A B ( ) ( ) ( )cP A B P A P A B
( ) ( ) ( )cP B P A B P A B ( ) ( ) ( )cP A B P B P A B ( ) ( ) ( ) ( )P A B P A P B P A B
More on Theorem 1-2
A
B C
?( )P A B C
( ) ( ) ( )P A P B P C
( ) ( ) ( )P A B P B C P C A
( )P A B C
Theorem 1-3
1 21
( )n
i ni
P A P A A A
1
( )n
ii
P A
1
( )i ji j n
P A A
1
( )i j ki j k n
P A A A
1
1 2( 1) ( )nnP A A A
See the text for the proof.
Theorem 1-3
1 21
( )n
i ni
P A P A A A
1
( )n
ii
P A
1
( )i ji j n
P A A
1
( )i j ki j k n
P A A A
1
1 2( 1) ( )nnP A A A
S1
S2
S3
Sn
11 2 3
1
( 1)n
ni n
i
P A S S S S
1
1
( 1)n
kk
k
S
Example 12
Let A, B be two events of a sample space with P(A) = 1/2 , P(B) = 1/2, P(A B) = 2/3
1. Find the probabilities of the following events:(1) P(Ac) (2) P(Bc)
(3) P(A B) (4) P(Ac B)
(5) P(Ac B) (6) P(A Bc)
(7) P(Ac Bc) (8) P(Ac Bc).
2. Are A, B mutually exclusive?
A B1/3
1/61/6
Chapter 1Probability Spaces
Conditional Probabilities
Definition
Let A, B be two events such that P(A) > 0. Then the conditional probability of B given A, written P(B|A), is defined by
( )( | )
( )
P A BP B A
P A
A BA B
If P(A) = 0, then P(B|A) is undefined.
Example 13
Toss three balanced coins. Let A denote the event that two coins land heads, B denote that the first coin lands heads. Find P(A), P(B), P(B|A), and P(A|B).
Toss three balanced coins. Let A denote the event that two coins land heads, B denote that the first coin lands heads. Find P(A), P(B), P(B|A), and P(A|B).
, , , , , , ,HHH HHT HTH HTT THH THT TTH TTT
, ,A HHT HTH THH
, , ,B HHH HHT HTH HTT
,A B HHT HTH
( ) 3 / 8P A
( ) 1/ 2P B ( ) 1/ 4P A B
( ) 1/ 4 2( | )
( ) 3 / 8 3
P A BP B A
P A
( ) 1/ 4 1( | )
( ) 1/ 2 2
P A BP A B
P B
By Product
( )( ) 0
( )( | )
( ) 0
P A BP A
P AP B A
undefined P A
( )( ) 0
( )( | )
( ) 0
P A BP B
P BP A B
undefined P B
( | ) ( ) ( ) 0
( ) ( | ) ( ) ( ) 0
0
P A B P B P B
P A B P B A P A P A
otherwise
Chapter 1Probability Spaces
Independence of Events
Definition
Two events A and B are independent if and only if
( ) ( ) ( )P A B P A P B
A B
Example 14
Toss one balanced coin three times. Let A denote
the event that the first two tosses land heads, B
denote that exactly one toss lands heads, and C
denote that the third toss lands tails.
1. Are A, B independent?
2. Are A, C independent?
3. Are B, C independent?
Toss one balanced coin three times. Let A denote
the event that the first two tosses land heads, B
denote that exactly one toss lands heads, and C
denote that the third toss lands tails.
1. Are A, B independent?
2. Are A, C independent?
3. Are B, C independent?
Example 14
Toss one balanced coin three times. Let A denote the event that the first two tosses land heads, B denote that exactly one toss lands heads, and C denote that the third toss lands tails.1. Are A, B independent?2. Are A, C independent?3. Are B, C independent?
Toss one balanced coin three times. Let A denote the event that the first two tosses land heads, B denote that exactly one toss lands heads, and C denote that the third toss lands tails.1. Are A, B independent?2. Are A, C independent?3. Are B, C independent?
, , , , , , ,HHH HHT HTH HTT THH THT TTH TTT
,A HHH HHT
, ,B HTT THT TTH
, , ,C HHT HTT THT TTT
A B
{ }A C HHT
,B C HTT THT
( ) 1/ 4P A
( ) 3 / 8P B
( ) 1/ 2P C
( ) 0P A B
( ) 1/ 8P A C
( ) 1/ 4P B C
( ) ( )P A P B
( ) ( )P A P C
( ) ( )P B P C
Theorem 1-4
A BAc BA Bc
Ac Bc
Theorem 1-5
A B P(B|A)=P(B)
( )( | )
( )
P A BP B A
P A
( ) ( )
( )
P A P B
P APf)
Definition
n events A1, A2, . . . , An are said to be
mutually independent if and only if for any
different
k = 2, . . . , n events satisfy
1 2, , , kA A A
1 2 1 2( ) ( ) ( ) ( )k kP A A A P A P A P A
Example 15
Given the following Venn Diagram, are events A,
B, C mutually independent?
A
C
B0.06 0.24 0.18
0.060.24
0.14
0.06
A
C
B0.06 0.24 0.18
0.060.24
0.14
0.06
P(A) = 0.6, P(B) = 0.8, P(C) = 0.5
P(ABC) = 0.24 = P(A)P(B)P(C)
P(AB) = 0.48 = P(A)P(B)
P(AC) = 0.30 = P(A)P(C)
P(BC) = 0.38 P(B)P(C)
Remarks
1. P(A1 A2 · · · An) = P(A1)P(A2) · · · P(An) does not imply that A1, A2, . . . , An are pairwise independent (Example 15).
2. A1, A2, . . . , An are pairwise independent does not imply that they are mutually independent.
Example 16
Toss two dice. Let
A = “The 1st die is 1,2, or 3”
B = “The 2nd die is 4,5, or 6”
C = “The sum of two dice is 7”
Show that A, B, C are pairwise independent but not m
utually independent.
Toss two dice. Let
A = “The 1st die is 1,2, or 3”
B = “The 2nd die is 4,5, or 6”
C = “The sum of two dice is 7”
Show that A, B, C are pairwise independent but not m
utually independent.
Chapter 1Probability Spaces
Reliabilities
Reliabilities
Reliability of a component Let Ri denote the probability of a component in a system which is functioning properly (event Ai).
We will assume that the failure events of components in a system are mutually independent.
Reliabilities of systems 1. series systems — The entire system will fail if any one of its component fails.2. parallel systems — The entire system will fail only if all its component fail.
Reliability of a Series System (Rss)
C1C1 C2
C2 CnCnA B
("The series system functions properly")ssR P
1 2( )nP A A A
1 2( ) ( ) ( )nP A P A P A
1
n
ii
R
Reliability of a Parallel System (Rps)
C1C1
C2C2
CnCn
A B...
("The parallel system functions properly")psR P
1 21 ( )c c cnP A A A
1
1 (1 )n
ii
R
1 21 ( ) ( ) ( )c c cnP A P A P A
1 21 [1 ( )][1 ( )] [1 ( )]nP A P A P A
Example 17
Consider the following system.
C1C1
C2C2
C3C3A B
Let R1 = R2 = R3 = 0.95.Find the reliability of the system.
Example 17
Consider the f ollowing system.
C1C1
C2C2
C3C3A B
Let R1 = R2 = R3 = 0.95.Find the reliability of the system.
Consider the f ollowing system.
C1C1
C2C2
C3C3A B
Let R1 = R2 = R3 = 0.95.Find the reliability of the system.
Consider the f ollowing system.
C1C1
C2C2
C3C3A B
Let R1 = R2 = R3 = 0.95.Find the reliability of the system.
Consider the f ollowing system.
C1C1
C2C2
C3C3A B
Let R1 = R2 = R3 = 0.95.Find the reliability of the system.
1 2 3( )R P A A A
1 3 2 3( ) ( )P A A A A
1 3 2 3 1 3 2 3( ) ( ) ( ) ( )P A A P A A P A A A A
1 3 2 3 1 2 3( ) ( ) ( )P A A P A A P A A A
1 3 2 3 1 2 3( ) ( ) ( ) ( ) ( ) ( ) ( )P A P A P A P A P A P A P A
1 3 2 3 1 2 3R R R R R R R 2 2 30.95 0.95 0.95
0.947625
Chapter 1Probability Spaces
Bayes’ Rule
A Story The umbrella must have been made in Taiwan
Event Space A Partition of Sample Space
日 德 法
義台
B1 B2
B3
B4
B5
, i jB B i j
1
n
ii
B
1{ , , }nB B a partition of
Event Interests Us (A)
日 德 法
義台
B1 B2
B3
B4
B5
, i jB B i j
1
n
ii
B
1{ , , }nB B a partition of
A
Preliminaries
日 德 法
義台
B1 B2
B3
B4
B5 APrior Probabilities:
( ), 1, ,iP B i n
Likelihoods:
( | ), 1, ,iP A B i n
Law of Total Probability
日 德 法
義台
B1 B2
B3
B4
B5 APrior Probabilities:
( ), 1, ,iP B i n
Likelihoods:
( | ), 1, ,iP A B i n
1 1( ) ( | ) ( ) ( | ) ( )n nP A P A B P B P A B P B
Law of Total Probability
日 德 法
義台
B1 B2
B3
B4
B5 APrior Probabilities:
( ), 1, ,iP B i n
Likelihoods:
( | ), 1, ,iP A B i n
1
( ) ( | ) ( )n
j jj
P A P A B P B
Goal: Posterior Probabilities
日 德 法
義台
B1 B2
B3
B4
B5 APrior Probabilities:
( ), 1, ,iP B i n
Likelihoods:
( | ), 1, ,iP A B i n
1
( ) ( | ) ( )n
j jj
P A P A B P B
Goal: ( | ) ,? 1, ,iP B A i n
Goal: Posterior Probabilities
Prior Probabilities:
( ), 1, ,iP B i n
Likelihoods:
( | ), 1, ,iP A B i n
1
( ) ( | ) ( )n
j jj
P A P A B P B
Goal: ( | ) ,? 1, ,iP B A i n
( )( | )
( )i
i
P A BP B A
P A
( | ) ( )
( )i iP A B P B
P A
Bayes’ Rule
Given( ), 1, ,iP B i n ( | ), 1, ,iP A B i n
1
( ) ( | ) ( )n
j jj
P A P A B P B
( | ) ( )( | ) , 1, ,
( )i i
i
P A B P BP B A i n
P A
Example 18
Suppose that the population of a certain city is 40% male and 60%
female. Suppose also that 50% of the males and 30% of the females
smoke. Find the probability that a smoker is male.
Suppose that the population of a certain city is 40% male and 60%
female. Suppose also that 50% of the males and 30% of the females
smoke. Find the probability that a smoker is male.
M : A selected person is maleW : A selected person is femaleS : A selected person who smokes
Define
( ) 0.4P M
We are givenWe are given
( | ) ( )( | )
( )
P S M P MP M S
P S
( ) ( | ) ( ) ( | ) ( )P S P S M P M P S W P W 0.5 0.4 0.3 0.6 0.38
0.5 0.4
0.38
0.526
?( | )P M S
( ) 0.6P W ( | ) 0.5P S M ( | ) 0.3P S W
M
W
S
NS
S
NS
0.4
0.6
0.5
0.5
0.3
0.7
Example 19
Consider a binary communication channel. Owing to noise, error
may occur.
For a given channel, assume a probability of 0.94 that a transmitted
0 is correctly received as a 0 and a probability of 0.91 that a
transmitted 1 is received as a 1. Further assumed a probability of
0.45 of transmitting of a 0. Determine
1. Probability that a 1 is received.
2. Probability that a 0 is received.
3. Probability that a 1 was transmitted, given that a 1 was received.
4. Probability that a 0 was transmitted, given that a 0 was received.
5. Probability of an error.
Consider a binary communication channel. Owing to noise, error
may occur.
For a given channel, assume a probability of 0.94 that a transmitted
0 is correctly received as a 0 and a probability of 0.91 that a
transmitted 1 is received as a 1. Further assumed a probability of
0.45 of transmitting of a 0. Determine
1. Probability that a 1 is received.
2. Probability that a 0 is received.
3. Probability that a 1 was transmitted, given that a 1 was received.
4. Probability that a 0 was transmitted, given that a 0 was received.
5. Probability of an error.
Example 19
Consider a binary communication channel. Owing to noise, error
may occur.
For a given channel, assume a probability of 0.94 that a transmitted
0 is correctly received as a 0 and a probability of 0.91 that a
transmitted 1 is received as a 1. Further assumed a probability of
0.45 of transmitting of a 0. Determine
1. Probability that a 1 is received.
2. Probability that a 0 is received.
3. Probability that a 1 was transmitted, given that a 1 was received.
4. Probability that a 0 was transmitted, given that a 0 was received.
5. Probability of an error.
Consider a binary communication channel. Owing to noise, error
may occur.
For a given channel, assume a probability of 0.94 that a transmitted
0 is correctly received as a 0 and a probability of 0.91 that a
transmitted 1 is received as a 1. Further assumed a probability of
0.45 of transmitting of a 0. Determine
1. Probability that a 1 is received.
2. Probability that a 0 is received.
3. Probability that a 1 was transmitted, given that a 1 was received.
4. Probability that a 0 was transmitted, given that a 0 was received.
5. Probability of an error.
0
1
0
1
0.94
0.06
0.91
0.09
0.45
0.55
T0 : A 0 is transmittedT1 : A 1 is transmittedR0 : A 0 is receivedR1 : A 1 is received
Define
T0 R0
T1 R1
1( ) ?P R
0( ) ?P R 1 1 ?( | )P T R
0 0 ?( | )P T R ?(" ")P Error
Example 19
0
1
0
1
0.94
0.06
0.91
0.09
0.45
0.55
T0 R0
T1 R1
1 ?1. ( )P R
0 ?2. ( )P R
1 13. ( | ) ?P T R
0 04. ( | ) ?P T R
5. (" ") ?P Error
1 1 1 1 0 0( | ) ( ) ( | ) ( )P R T P T P R T P T 0.91 0.55 0.06 0.45 0.5275
1 0.5275 0.4725
1 1 1
1
( | ) ( )
( )
P R T P T
P R0.91 0.55
0.5275
0.9488
0 0 0
0
( | ) ( )
( )
P R T P T
P R0.94 0.45
0.4725
0.8952
0 1 1 0( ) ( )P T R P T R 1 0 0 0 1 1( | ) ( ) ( | ) ( )P R T P T P R T P T
0.06 0.45 0.09 0.55 0.0765
Example 20
There are n boxes each contains a white and b black balls. Now randomly select one ball from the 1st box to put it into the 2nd box, then randomly select one ball from the 2nd box to put it into the 3r
d box, . . . This procedure is continued. Let Wi and Bi denote the ev
ents that the chosen ball from the ith box being white and black, respectively. Determine
1. P(Wn) = ?;
2. P(Wn|W1) = ?;
3. limn→∞P(Wn|W1) = ?
There are n boxes each contains a white and b black balls. Now randomly select one ball from the 1st box to put it into the 2nd box, then randomly select one ball from the 2nd box to put it into the 3r
d box, . . . This procedure is continued. Let Wi and Bi denote the ev
ents that the chosen ball from the ith box being white and black, respectively. Determine
1. P(Wn) = ?;
2. P(Wn|W1) = ?;
3. limn→∞P(Wn|W1) = ?
Example 20
There are n boxes each contains a white and b black balls. Now randomly select one ball from the 1st box to put it into the 2nd box, then randomly select one ball from the 2nd box to put it into the 3r
d box, . . . This procedure is continued. Let Wi and Bi denote the ev
ents that the chosen ball from the ith box being white and black, respectively. Determine
1. P(Wn) = ?;
2. P(Wn|W1) = ?;
3. limn→∞P(Wn|W1) = ?
There are n boxes each contains a white and b black balls. Now randomly select one ball from the 1st box to put it into the 2nd box, then randomly select one ball from the 2nd box to put it into the 3r
d box, . . . This procedure is continued. Let Wi and Bi denote the ev
ents that the chosen ball from the ith box being white and black, respectively. Determine
1. P(Wn) = ?;
2. P(Wn|W1) = ?;
3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n1 n
. . .
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
1 1 1 1( ) ( | ) ( ) ( | ) ( )n n n n n n nP W P W W P W P W B P B
1( )a
P Wa b
1
1
a
a b
1
a
a b
Define pi=P(Wi).
1np 11 np
1p
np
1
1
1
1 1n n
ap
a ba
p pa b a b
1
1
1
1 1n n
ap
a ba
p pa b a b
np
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
1
1
1
1 1n n
ap
a ba
p pa b a b
1
1
1
1 1n n
ap
a ba
p pa b a b
np
1n np cp d
1 2n np cp d
2 3n np cp d
3 2p cp d
2 1p cp d
.
.
.
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
1
1
1
1 1n n
ap
a ba
p pa b a b
1
1
1
1 1n n
ap
a ba
p pa b a b
np
1n np cp d
1 2n np cp d
2 3n np cp d
3 2p cp d
2 1p cp d
.
.
.
21 2n ncp c p cd
2 3 22 3n nc p c p c d
3 2 33 2
n n nc p c p c d 2 1 2
2 1n n nc p c p c d
11
1
(1 )
1
nn
n
d cp c p
c
21 2n ncp c p cd
2 3 22 3n nc p c p c d
3 2 33 2
n n nc p c p c d 2 1 2
2 1n n nc p c p c d
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
1
1
1
1 1n n
ap
a ba
p pa b a b
1
1
1
1 1n n
ap
a ba
p pa b a b
np
1n np cp d
.
.
.
11
1
(1 )
1
nn
n
d cp c p
c
1
1
11
1 1111 1
1
n
n
n
aa b a ba
pa b a b
a b
1
1
11
1 1111 1
1
n
n
n
aa b a ba
pa b a b
a b
a
a b
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
1
1
1
1 1n n
ap
a ba
p pa b a b
1
1
1
1 1n n
ap
a ba
p pa b a b
np
1
1
11
1 1111 1
1
n
n
n
aa b a ba
pa b a b
a b
a
a b
11
1
(1 )
1
nn
n
d cp c p
c
1
1( | )nP W W
11
1
nb
a b a b
1
1
1( | )
1
n
n
b aP W W
a b a b a b
Example 20 1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
1. P(Wn) = ?;2. P(Wn|W1) = ?;3. limn→∞P(Wn|W1) = ?
ab
ab
ab
ab
ab
1 2 3 n 1 n
. . .abab
abab
abab
abab
abab
1 2 3 n 1 n
. . .
np
1
1
1( | )
1
n
n
b aP W W
a b a b a b
1lim ( | )nn
aP W W
a b
( )nP W