Post on 01-Feb-2016
description
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USF, Math & Stat
Introduction to Probability
STA 4442.001: Fall 2015
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USF, Math & Stat
Instructor: Dan Shen
Office: CMC 324
Phone: (813)974-5062
Email: danshen@usf.edu
Instructor Information
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Office hours: MW 9:45-10:45 a.m
Location: CMC 324
Office Hour
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“Probability for Engineering, Mathematics and Science’’
By Chris P. Tsokos,
Brooks/Cole (ISBN-13:978-1-111-43027-6).
Textbook
• Bring the textbook and a calculator to every class meeting.
• Do not bring laptop computers to class.
• Please turn off your cell phone during class time.
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• Use your USER ID and PASSWORD to login canvas
• Homework assignments and important announcements will be posted there.
Canvas
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• Assigned in each class. Monday home due day is next Monday and so on.
Necessary adjustments will be made right before each exam.
• No late homeworks will be accepted.
• Homeworks will not be accepted via email, disk, or any other electronic form.
• Missed homeworks will receive a grade of zero.
Homework
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• Show all work neatly, write in blue or black pen or pencil (never in red);
• Clearly label each problem, circle your numerical answers; • Staple the entire assignment together in the correct order (that is, the order in which problems were assigned.) with your name printed (in blue or black ink) on every page. Any homework violating any of these rules will receive a grade of zero for the entire assignment. Check your homework grades in “Canvas” after your homework is returned.
Homework
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• Homeworks, total 20% of your course grade; The lowest one will be dropped
• 6 Quizzes, total 15% of your course grade;
The lowest one will be dropped
• 3 in-class Midterm exams, total 45% of your course grade;
The lowest one will be dropped
• Final Exam, 20% of your course grade; • No make-up exams. Missed exams will receive a grade of zero; • Closed-book and closed-note with no formula sheets permitted; • Computers are not permitted during exams, but calculators may be used.
Grading
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•Drop/Add ends, fee liability/tuition payment deadline: Friday, August 28
•Last day to drop with a "W"; no refund & no academic penalty: Saturday, October 31
Drop the class
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• Feel free to approach the instructor with any concerns you may have
regarding the course
• Each student is responsible for verifying his or her recorded scores
(homeworks & midterm exams), which will be posted on canvas,
during the semester.
• The Honor Code will be observed at all times in this course.
• This class will participate in the Course Evaluation.
Course Concern
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Probability
Example 1.1.1 Tossing a fair die
Probability of obtaining an odd number?
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Example 1.1.1 Tossing a fair die
sample space: S={x, x=1, 2, 3, 4, 5, 6}
sample space, sample point, and sample event
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Example 1.1.1 Tossing a fair die
sample space: S={x, x=1, 2, 3, 4, 5, 6}
sample point: for example, x=1
sample space, sample point, and sample event
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample space: S={x, x=1, 2, 3, 4, 5, 6}
sample point: for example, x=1
sample event: obtaining an odd number S1={x, x=1, 3, 5}
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Sample space, point, event
Example: flip a coin twice
H: head
T: tail
H
T
First flip
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Sample space, point, event
Example: flip a coin twice
H: head
T: tail
H
T
First flip
H
T
H
T
Second flip
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space:
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
sample point: for example, HH
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
sample point: for example, HH
sample event 1: the fist and second flip are both heards
S1={HH}
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
sample point: for example, HH
sample event 1: the fist and second flip are both heards
S1={HH} a simple event
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
sample point: for example, HH
sample event 2: the fist flip is head
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Sample space, point, event
H
T
First flip
H
T H
T
Second flip outcomes
HH
HT TH
TT
sample space: S={HH, HT, TH, TT}
sample point: for example, HH
sample event 2: the fist flip is head
S2={HH, HT} a compound event
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Definitions
Discrete space S:
1. S contains a finite number of points
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Definitions
Discrete space S:
1. S contains a finite number of points
Example, S={x, x=1, 1.5, 2, 2.5}
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Definitions
Discrete space S:
1. S contains a finite number of points
2. S contains an infinite number of points that can be
put into a one to one correspondence with the
positive integer
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Definitions
Discrete space S:
1. S contains a finite number of points
2. S contains an infinite number of points that can be
put into a one to one correspondence with the
positive integer
Example, S={x, x=1, 1.5, 2, 2.5, ……}
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Definitions
Discrete space S:
1. S contains a finite number of points
2. S contains an infinite number of points that can be
put into a one to one correspondence with the
positive integer
Continuous space S: S contains a continuum of points
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Definitions
Discrete space S:
1. S contains a finite number of points
2. S contains an infinite number of points that can be
put into a one to one correspondence with the
positive integer
Continuous space S: S contains a continuum of points
Examples, S={t, 0≤ t< +∞}
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Definitions
Discrete space S:
1. S contains a finite number of points
2. S contains an infinite number of points that can be
put into a one to one correspondence with the
positive integer
Continuous space S: S contains a continuum of points
S={t, 0< t< 1} is continuous space?????
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Definitions
Two events S1= S2: if they contain same points.
Impossible (empty) event S1 denoted by Ø :
• S1 contains no sample point
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Definitions
Two events S1= S2: if they contain same points.
Impossible (empty) event S1 denoted by Ø :
• S1 contains no sample point
For example S1 ={x, x=7, 8}
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Definitions
Complement event of S1 denoted by S-S1 or S1 :
• Event contains sample points in S but not in S1
_
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1=???
_
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞}
_
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2=???
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
_ _
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Definitions
Complement event of S1 denoted by S-S1 or S1 :
• Event contains sample points in S but not in S1
Union of S1 and S2 denoted by S1 ∪ S2 :
• Event contains all sample points in S1 and S2
_
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2=???
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2={t; 25< t≤140 }
_ _
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Definitions
Complement event of S1 denoted by S-S1 or S1 :
• Event contains sample points in S but not in S1
Union of S1 and S2 denoted by S1 ∪ S2 :
• Event contains all sample points in S1 and S2
Intersection of S1 and S2 denoted by S1 ∩ S2 :
• Event contains sample points in both S1 and S2
_
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2={t; 25< t≤140 } S1 ∩ S2=???
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 =???
_ _
____
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
_ _
____
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 =???
_ _
____
____
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
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____
____
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
S1 ∪ S2 =???
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____
____
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
S1 ∪ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
_ _
____
____
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
S1 ∪ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∩ S2 =???
_ _
____
____
_ _
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
S1 ∪ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∩ S2 ={t; 0≤ t ≤25 or 140< t<+∞ }
_ _
____
____
_ _
_ _
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Example
S={t; 0≤ t <+∞} S1={t; 25< t≤100 } S2={t; 60< t≤140 }
S1={t; 0≤ t ≤25 or 100<t <+∞} S2={t; 0≤ t ≤60 or 140<t <+∞}
S1 ∪ S2 ={t; 25< t≤140 } S1 ∩ S2={t; 60< t≤100 }
S1 ∩ S2 ={t; 0≤ t ≤60 or 100<t <+∞}
S1 ∪ S2 ={t; 0≤ t ≤25 or 140< t<+∞ } S1 ∩ S2 = S1 ∪ S2
S1 ∪ S2 ={t; 0≤ t ≤60 or 100<t <+∞} S1 ∪ S2 = S1 ∩ S2
S1 ∩ S2 ={t; 0≤ t ≤25 or 140< t<+∞ } De Morgan’s laws
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____
____
_ _
_ _
____ _ _
_ _ ____
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Definitions
Complement event of S1 denoted by S-S1 or S1 :
• Event contains sample points in S but not in S1
Union of S1 and S2 denoted by S1 ∪ S2 :
• Event contains all sample points in S1 and S2
Intersection of S1 and S2 denoted by S1 ∩ S2 :
• Event contains sample points in both S1 and S2
S1 and S2 are mutually exclusive events or disjoint events • S1 ∩ S2= Ø
_
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
S1 ∩ S2= Ø
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
Pr( S1)=???
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
Pr( S1)=3/6=1/2
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
0 ≤ Pr( S1) ≤ 1
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
sample event 1: obtaining an odd number S1={x, x=1, 3, 5}
sample event 2: obtaining an even number S2={x, x=2, 4, 6}
0 ≤ Pr( S1) ≤ 1
0 ≤ Pr( S2) ≤ 1
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Axioms
Axiom 1.2.1 0 ≤ Pr( Si) ≤ 1
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
S={x, x=1, 2, 3,4, 5,6}
Pr( S)= ???
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
S={x, x=1, 2, 3,4, 5,6}
Pr( S)= 1
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Axioms
Axiom 1.2.1 0 ≤ Pr( Si) ≤ 1
Axiom 1.2.2 Pr( S) = 1
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= ???
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= ??? for i≠j
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
S1 ∪ S2 ∪ S3= ???
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
S1 ∪ S2 ∪ S3= {1, 2, 3}
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
S1 ∪ S2 ∪ S3= {1, 2, 3}
Pr(S1 ∪ S2 ∪ S3 ) =???
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
S1 ∪ S2 ∪ S3= {1, 2, 3}
Pr(S1 ∪ S2 ∪ S3 ) =3/6
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sample space, sample point, and sample event
Example 1.1.1 Tossing a fair die
Si ={ i}, i=1, …, 6
Pr(Si )= 1/6
Si ∩ Sj= Ø for i≠j
S1 ∪ S2 ∪ S3= {1, 2, 3}
Pr(S1 ∪ S2 ∪ S3 ) =3/6= Pr(S1 )+ Pr(S2 )+ Pr( S3 )
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Axioms
Axiom 1.2.1 0 ≤ Pr( Si) ≤ 1
Axiom 1.2.2 Pr( S) = 1
Axiom 1.2.3 Si ∩ Sj= Ø for i≠j =1, 2, 3, …, n, ….
Pr(S1 ∪ S2… ∪ Sn ∪ … ) = Pr(S1 )+Pr( S2)+…+Pr(Sn)+….
or
11
)Pr()Pr(i
iii
SS
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
21 SS
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
21 SS
S1
S2
S
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region
21 SS
S1
S2
S
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region= S1 ∪??
21 SS
S1
S2
S
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region= S1 ∪(S2 ∩ S1 )
21 SS
S1
S2
S
_
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region= S1 ∪(S2 ∩ S1 )
From Axiom 1.2.3, Pr(S2)= Pr(S1)+ Pr(S2 ∩ S1 )
21 SS
S1
S2
S
_
_
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region= S1 ∪(S2 ∩ S1 )
From Axiom 1.2.3, Pr(S2)= Pr(S1)+ Pr(S2 ∩ S1 )
From Axiom 1.2.1, Pr(S2 ∩ S1 ) ≥ 0
21 SS
S1
S2
S
_
_
_
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Theorems
Theorem 1.2.1 two events then Pr( S1) ≤ Pr( S2)
Proof S2= blue region + red region= S1 ∪(S2 ∩ S1 )
From Axiom 1.2.3, Pr(S2)= Pr(S1)+ Pr(S2 ∩ S1 )
From Axiom 1.2.1, Pr(S2 ∩ S1 ) ≥ 0
Thus Pr(S1)≤ Pr(S2)
21 SS
S1
S2
S
_
_
_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Sk
S
_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Proof S= red region + white region
Sk
S
_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Proof S= red region + white region = Sk ∪( Sk )
Sk
S
_
_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Proof S= red region + white region = Sk ∪( Sk )
From Axiom 1.2.3, Pr(S)= Pr(Sk)+ Pr(Sk )
Sk
S
_
_
_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Proof S= red region + white region = Sk ∪( Sk )
From Axiom 1.2.3, Pr(S)= Pr(Sk)+ Pr(Sk )
From Axiom 1.2.2, Pr(S)= 1
Sk
S
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_
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Theorems
Theorem 1.2.3 For any event Sk , we have Pr( Sk) =1- Pr( Sk)
Proof S= red region + white region = Sk ∪( Sk )
From Axiom 1.2.3, Pr(S)= Pr(Sk)+ Pr(Sk )
From Axiom 1.2.2, Pr(S)= 1
Then Pr(Sk)= 1- Pr(Sk )
Sk
S
_
_
_
_
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USF, Math & Stat
Theorems
Theorem 1.2.4 For impossible event Ø, we have Pr(Ø) =0
Proof From Theorem 1.2.3, Pr(S)= 1- Pr(S)
_
90
USF, Math & Stat
Theorems
Theorem 1.2.4 For impossible event Ø, we have Pr(Ø) =0
Proof From Theorem 1.2.3, Pr(S)= 1- Pr(S)
Note that Pr(S)= 1 and Ø= S
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Theorems
Theorem 1.2.4 For impossible event Ø, we have Pr(Ø) =0
Proof From Theorem 1.2.3, Pr(S)= 1- Pr(S)
Note that Pr(S)= 1 and Ø= S
Then Pr(Ø)= 0
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Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
S2 ∩ S1 S1 S2
93
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2 ∩ S1 S1 S2
94
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2= red + blue
S2 ∩ S1 S1 S2
95
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2= red + blue
S1 ∪ S2= yellow+blue+red
S2 ∩ S1 S1 S2
96
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2= red + blue
S1 ∪ S2= yellow+blue+red
S1 ∩ S2 =blue
S2 ∩ S1 S1 S2
97
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2= red + blue
S1 ∪ S2= yellow+blue+red
S1 ∩ S2 =blue
then Pr(S1)+Pr(S2) = Pr(S1 ∪ S2) + Pr(S1 ∩ S2 )
S2 ∩ S1 S1 S2
98
USF, Math & Stat
Theorems
Theorem 1.2.5 For two events S1 and S2,
Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
Proof :
S1= yellow + blue
S2= red + blue
S1 ∪ S2= yellow+blue+red
S1 ∩ S2 =blue
then Pr(S1)+Pr(S2) = Pr(S1 ∪ S2) + Pr(S1 ∩ S2 )
It follows that Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )
S2 ∩ S1 S1 S2
99
USF, Math & Stat
Theorems
Theorem 1.2.6 For a sequence of events S1, …., Sn
)...Pr()1(... 21
1
n
n SSS
n
kjikji
ji
n
jiji
ji
n
i
ii
n
i
SSSSSS,,1,11
)Pr()Pr()Pr()Pr(
100
USF, Math & Stat
Theorems
Theorem 1.2.6 For a sequence of events S1, …., Sn
If the events are disjoint, then
11
)Pr()Pr(i
iii
SS
)...Pr()1(... 21
1
n
n SSS
n
kjikji
ji
n
jiji
ji
n
i
ii
n
i
SSSSSS,,1,11
)Pr()Pr()Pr()Pr(