chapter 1.1-1.2 prob tso

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Introduction to Probability

STA 4442.001: Fall 2015

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Instructor: Dan Shen

Office: CMC 324

Phone: (813)974-5062

Email: [email protected]

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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sample space: S={x, x=1, 2, 3, 4, 5, 6}

sample space, sample point, and sample event

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sample space: S={x, x=1, 2, 3, 4, 5, 6}

sample point: for example, x=1


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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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Example: flip a coin twice

H: head

T: tail

H

T

First flip

H

T

H

T

Second flip

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H

T

First flip

H

T H

T

Second flip outcomes

HH

HT TH

TT

sample space:

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H

T

First flip

H

T H

T


HH

HT TH

TT

sample space: S={HH, HT, TH, TT}

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H

T

First flip

H

T H

T


HH

HT TH

TT


sample point: for example, HH

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H

T

First flip

H

T H

T


HH

HT TH

TT



sample event 1: the fist and second flip are both heards

S1={HH}

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H

T

First flip

H

T H

T


HH

HT TH

TT



sample event 1: the fist and second flip are both heards

S1={HH} a simple event

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H

T

First flip

H

T H

T


HH

HT TH

TT



sample event 2: the fist flip is head

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H

T

First flip

H

T H

T


HH

HT TH

TT



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:


Example, S={x, x=1, 1.5, 2, 2.5}

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Definitions

Discrete space S:


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:




positive integer

Example, S={x, x=1, 1.5, 2, 2.5, ……}

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Definitions

Discrete space S:




positive integer

Continuous space S: S contains a continuum of points

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Definitions

Discrete space S:




positive integer


Examples, S={t, 0≤ t< +∞}

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Definitions

Discrete space S:




positive integer


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



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





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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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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____

____

_ _

_ _

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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





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 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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S1 ∩ S2= Ø

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Pr( S1)=???

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Pr( S1)=3/6=1/2

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0 ≤ Pr( S1) ≤ 1

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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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S={x, x=1, 2, 3,4, 5,6}

Pr( S)= ???

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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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Si ={ i}, i=1, …, 6

Pr(Si )= ???

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6

Si ∩ Sj= ??? for i≠j

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6

Si ∩ Sj= Ø for i≠j

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6


S1 ∪ S2 ∪ S3= ???

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6


S1 ∪ S2 ∪ S3= {1, 2, 3}

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6


S1 ∪ S2 ∪ S3= {1, 2, 3}

Pr(S1 ∪ S2 ∪ S3 ) =???

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6


S1 ∪ S2 ∪ S3= {1, 2, 3}

Pr(S1 ∪ S2 ∪ S3 ) =3/6

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Si ={ i}, i=1, …, 6

Pr(Si )= 1/6


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


21 SS

S1

S2

S

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Theorems


Proof S2= blue region + red region

21 SS

S1

S2

S

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Theorems


Proof S2= blue region + red region= S1 ∪??

21 SS

S1

S2

S

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Theorems


Proof S2= blue region + red region= S1 ∪(S2 ∩ S1 )

21 SS

S1

S2

S

_

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Theorems



From Axiom 1.2.3, Pr(S2)= Pr(S1)+ Pr(S2 ∩ S1 )

21 SS

S1

S2

S

_

_

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Theorems




From Axiom 1.2.1, Pr(S2 ∩ S1 ) ≥ 0

21 SS

S1

S2

S

_

_

_

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Theorems




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


Proof S= red region + white region

Sk

S

_

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Theorems


Proof S= red region + white region = Sk ∪( Sk )

Sk

S

_

_

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Theorems



From Axiom 1.2.3, Pr(S)= Pr(Sk)+ Pr(Sk )

Sk

S

_

_

_

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Theorems




From Axiom 1.2.2, Pr(S)= 1

Sk

S

_

_

_

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Theorems




From Axiom 1.2.2, Pr(S)= 1

Then Pr(Sk)= 1- Pr(Sk )

Sk

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)

_

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Theorems



Note that Pr(S)= 1 and Ø= S

_

_

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Theorems



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

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Theorems


Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )

Proof :

S1= yellow + blue

S2 ∩ S1 S1 S2

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Theorems


Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )

Proof :

S1= yellow + blue

S2= red + blue

S2 ∩ S1 S1 S2

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Theorems


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

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Theorems


Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )

Proof :

S1= yellow + blue

S2= red + blue


S1 ∩ S2 =blue

S2 ∩ S1 S1 S2

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Theorems


Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )

Proof :

S1= yellow + blue

S2= red + blue


S1 ∩ S2 =blue

then Pr(S1)+Pr(S2) = Pr(S1 ∪ S2) + Pr(S1 ∩ S2 )

S2 ∩ S1 S1 S2

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Theorems


Pr(S1 ∪ S2)= Pr(S1) + Pr(S2) - Pr(S1 ∩ S2 )

Proof :

S1= yellow + blue

S2= red + blue


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

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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(

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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(

chapter 1.1-1.2 prob tso

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