Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo ([email protected])

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

Outline

1 Chapter 2ExamplesDefinition and illustrations


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


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


Outline

1 Chapter 2ExamplesDefinition and illustrations


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.



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,





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,





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,




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)



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)



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)



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)



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)



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)



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.






...so P(ω3) = 1/3.

Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.






...so P(ω3) = 1/3. Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.






...so P(ω3) = 1/3. Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.






...so P(ω3) = 1/3. Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.






...so P(ω3) = 1/3. Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.






...so P(ω3) = 1/3. Wrong!




Ω = ω1, ω2, ω3, ω4 = HH,TT ,HT ,TH.


A = HT ,TH.



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|




Outcomes: What is Ω and |Ω|?

Event: What is the event A ⊆ Ω of interest?


P(A) =|A||Ω|






P(A) =|A||Ω|






P(A) =|A||Ω|


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Probability Math 371 - Department of Mathematics | …williamdemeo/Math371-Summer2011/...Chapter 2 Probability Math 371 University of Hawai‘i at Manoa¯ Summer 2011 W. DeMeo ([email protected])