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Transcript of Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2...
![Page 1: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/1.jpg)
Chapter 2Probability
Math 371
University of Hawai‘i at Manoa
Summer 2011
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 1 / 8
![Page 2: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/2.jpg)
Outline
1 Chapter 2ExamplesDefinition and illustrations
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 2 / 8
![Page 3: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/3.jpg)
Examples of some important basic concepts
Example 1. bushel of applesproportion
P(A) =|A||Ω|
(2.1.1)
(2.1.3) and (2.1.4).
Example 3. toss of a “perfect” dieequally likely outcomeseventsmutually exclusive eventsrelative frequencylimiting frequency
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 3 / 8
![Page 4: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/4.jpg)
Examples of some important basic concepts
Example 1. bushel of applesproportion
P(A) =|A||Ω|
(2.1.1)
(2.1.3) and (2.1.4).
Example 3. toss of a “perfect” dieequally likely outcomeseventsmutually exclusive eventsrelative frequencylimiting frequency
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 3 / 8
![Page 5: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/5.jpg)
Outline
1 Chapter 2ExamplesDefinition and illustrations
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 4 / 8
![Page 6: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/6.jpg)
Definition of Probability Measure
functions
“probability” function, P; a function defined on sets.Definition of power set P(Ω) and examples.A probability measure is a function P : P(Ω)→ [0,1] satisfying,for all sets A,B ⊆ Ω,
0 ≤ P(A) ≤ 1;If A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B);P(Ω) = 1.
The probability of an event A is a number, denoted P(A), whereasthe function P itself is called a probability measure. The values ofP are the probabilities of various events.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8
![Page 7: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/7.jpg)
Definition of Probability Measure
functions“probability” function, P; a function defined on sets.Definition of power set P(Ω) and examples.
A probability measure is a function P : P(Ω)→ [0,1] satisfying,for all sets A,B ⊆ Ω,
0 ≤ P(A) ≤ 1;If A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B);P(Ω) = 1.
The probability of an event A is a number, denoted P(A), whereasthe function P itself is called a probability measure. The values ofP are the probabilities of various events.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8
![Page 8: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/8.jpg)
Definition of Probability Measure
functions“probability” function, P; a function defined on sets.Definition of power set P(Ω) and examples.A probability measure is a function P : P(Ω)→ [0,1] satisfying,for all sets A,B ⊆ Ω,
0 ≤ P(A) ≤ 1;If A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B);P(Ω) = 1.
The probability of an event A is a number, denoted P(A), whereasthe function P itself is called a probability measure. The values ofP are the probabilities of various events.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8
![Page 9: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/9.jpg)
Definition of Probability Measure
functions“probability” function, P; a function defined on sets.Definition of power set P(Ω) and examples.A probability measure is a function P : P(Ω)→ [0,1] satisfying,for all sets A,B ⊆ Ω,
0 ≤ P(A) ≤ 1;If A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B);P(Ω) = 1.
The probability of an event A is a number, denoted P(A), whereasthe function P itself is called a probability measure. The values ofP are the probabilities of various events.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8
![Page 10: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/10.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.
A subset A ⊆ Ω of these points represents an event.Conduct an experiment and observe the outcome ω1 ∈ Ω.If ω1 ∈ A, then we say “the event A has occurred.”How “likely” is the event A?If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 11: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/11.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.A subset A ⊆ Ω of these points represents an event.
Conduct an experiment and observe the outcome ω1 ∈ Ω.If ω1 ∈ A, then we say “the event A has occurred.”How “likely” is the event A?If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 12: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/12.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.A subset A ⊆ Ω of these points represents an event.Conduct an experiment and observe the outcome ω1 ∈ Ω.
If ω1 ∈ A, then we say “the event A has occurred.”How “likely” is the event A?If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 13: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/13.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.A subset A ⊆ Ω of these points represents an event.Conduct an experiment and observe the outcome ω1 ∈ Ω.If ω1 ∈ A, then we say “the event A has occurred.”
How “likely” is the event A?If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 14: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/14.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.A subset A ⊆ Ω of these points represents an event.Conduct an experiment and observe the outcome ω1 ∈ Ω.If ω1 ∈ A, then we say “the event A has occurred.”How “likely” is the event A?
If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 15: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/15.jpg)
Experiments, outcomes, and events
Each point ω ∈ Ω represents a possible outcome of an experiment.A subset A ⊆ Ω of these points represents an event.Conduct an experiment and observe the outcome ω1 ∈ Ω.If ω1 ∈ A, then we say “the event A has occurred.”How “likely” is the event A?If all outcomes ω ∈ Ω are equally likely, then
P(A) =|A||Ω|
(2.1.11)
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8
![Page 16: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/16.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 17: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/17.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3.
Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 18: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/18.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 19: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/19.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 20: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/20.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 21: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/21.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 22: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/22.jpg)
Experiments, outcomes, and events
Example 4. “favorable” outcomes and poor old D’Alembert.
Toss two identical coins. The sample space is...
Ω = ω1, ω2, ω3 = two heads, two tails, a head and a tail
...so P(ω3) = 1/3. Wrong!
Give D’Alembert a computer...
...he could quickly estimate P(ω3) ≈ 1/2. Extra credit!
The problem: ωi above are not equally likely outcomes.Instead, let
Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.
“A head and a tail” is an event, not an outcome:
A = HT ,TH.
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8
![Page 23: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/23.jpg)
Experiments, outcomes, and events
Example 5. Roll five dice.Find the probability they all show different faces.
Outcomes: What is Ω and |Ω|?Event: What is the event A ⊆ Ω of interest?
Probability: What is |A|, and what is the probability of A?
P(A) =|A||Ω|
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8
![Page 24: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/24.jpg)
Experiments, outcomes, and events
Example 5. Roll five dice.Find the probability they all show different faces.
Outcomes: What is Ω and |Ω|?
Event: What is the event A ⊆ Ω of interest?
Probability: What is |A|, and what is the probability of A?
P(A) =|A||Ω|
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8
![Page 25: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/25.jpg)
Experiments, outcomes, and events
Example 5. Roll five dice.Find the probability they all show different faces.
Outcomes: What is Ω and |Ω|?Event: What is the event A ⊆ Ω of interest?
Probability: What is |A|, and what is the probability of A?
P(A) =|A||Ω|
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8
![Page 26: Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo (williamdemeo@gmail.com)](https://reader031.fdocuments.us/reader031/viewer/2022022515/5aff209f7f8b9a994d8ff3e8/html5/thumbnails/26.jpg)
Experiments, outcomes, and events
Example 5. Roll five dice.Find the probability they all show different faces.
Outcomes: What is Ω and |Ω|?Event: What is the event A ⊆ Ω of interest?
Probability: What is |A|, and what is the probability of A?
P(A) =|A||Ω|
W. DeMeo ([email protected]) Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8