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Chapter 1 Introduction
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Use of probability in daily life
A.
B.
C.
D.
Lotto
Batting average in baseball
Election Poll (statistics)
Weather Forecast
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Why is Probability important
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•
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Many things change all the time
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Signal strength (voltage/current)
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Life of light bulb
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Time of Rain
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Traffic arrival
Probability can be used to establish mathematical model so that
these phenomena can be analyzed in a quantitative way.
Applications:
–
–
Signal variations in communication system (due to error occurrence)
Traffic congestion problem (when input will occur)
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Various definitions of probability
A.
B.
C.
D.
Relative-frequency definition
Classical definition
Probability as a measure of belief
Axiomatic definition
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A. Relative-frequency definition
The experiment under consideration is repeated n times.
If the event A occurs na times, then its probability P(A) is defined as the
na
limit of the relative frequency n of the occurrence of A.
n
P( A)  lim na
n
Ex. Toss a coin
Comment. 1. based on experimental ground.
natural
na
2. n is finite, limit of n cannot be obtained
This definition is a hypothesis about the existence of the limit.
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B. Classical definition
The probability P(A) of an event A is found a priori without actual
experimentation.
If N is the total number of possible outcomes of the experiment, and in
N a of these outcomes the event A occurs, then P( A)  Na
N
Ex1. Roll a die
Pr [even] = 36
Ex2. Roll a pair of dies
p = Pr [ Sum of two outcomes = 7] = ？
1.
There are 11 possible sums：2, 3, ….,12
p= 1
11
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2.
Count all possible pairs of numbers, not distinguishing between the first
and second die.
（1,1）
（1,2）
（1,3）（2,2）
（1,4）（2,3）
（1,5）（2,4）（3,3）
total 21
（1,6）（2,5）（3,4）
( 2,6 ) ( 3,5 ) ( 4,4 )
3
p = 21
( 3,6 ) ( 4,5 )
（4,6 ) ( 5,5 )
( 5,6 )
Note: possible outcomes are not equally likely.
（1,2）is more likely than（1,1)
( 6,6）
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3.
Count all pairs; distinguish between 1st and 2nd die. There are 36
possibilities, of which 6 have sum = 7
（3,4）（4,3）（5,2）（2,5）（1,6）（6,1）
p= 6
36
Improved version of the classical def：
N
P( A)  a , provided all N outcomes are equally likely .
N
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Criticism of the classical definition.
1. The definition is circular.
What is “equally likely”.
2. can be used only for a limited class of problems
Ex. unfair coin, Pr [5] = 1 , …
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3. make implicit use of the relative-freq. interpretation of probability.
Pr [5]  1
6
4. number of outcomes may be infinite.
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Ex. (Bertrand Paradox) In a circle C of radius r we draw at “random” a
cord AB. What is the prob. that the length l of this cord is greater
than the length of the side of an inscribed equilateral triangle？
Solution. a. The center M of the cord AB is a “random” point.
If it lies inside the circle C1 of radius r/2
then
C
l  3r
M
A
P
 r2
B
r/2
4 1
 r2 4
C1
r
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b. Fixed point A
If B lies on the arc DBE,
Then
D
l  3r
B
2 r
P 3 1
2 r 3
A
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c. Assume AB is perpendicular to the diameter FK.
l  3r
if the center M of AB lies between G and H.
F
r
2
G
M
A
P r 1
2r 2
B
H
K
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• The above three solutions correspond to three different
experiments.
• The example shows the difficulties associated with the
classical definition.
• It is meaningless to talk about the prob. of an event; the
underlying experiment must also be clearly specified.
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C. Probability as a measure of Belief that something might or
might not be true.
Ex.” It is probable that X is guilty ”.
•
It is a statement about a single event, and not about averages of
mass phenomena.
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This concept of probability is a form of inductive reasoning.
Induction: inference of a generalized conclusion from particular
instances
deduction: the deriving of a conclusion by reasoning inference in which
the conclusion about particulars follows necessarily from general or
universal premises.
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D.
Axiomatic Definition
The probability of an event A is a number P(A) assigned to this event.
This number obeys the following three postulates but is otherwise
unspecified:
I.
P(A) is positive: P( A)  0
II.
The probability of the certain event S equals 1 : P(S)=1
III.
If A and B are mutually exclusive, then
P  A B   P  A  P  B 
The resulting theory is based on these postulates and on nothing else.
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•
•
Axiom:
1.
a fundamental principle widely accepted on its
intrinsic merit
2.
a statement accepted as true as the basis for
argument or inference
Postulate:
1.
a hypothesis advanced as an essential
presupposition, condition, or premise of a train of
reasoning.
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