Chapter 2 Probability Math 371 University of Hawaii at M anoa - - PowerPoint PPT Presentation

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Chapter 2 Probability Math 371 University of Hawaii at M anoa - - PowerPoint PPT Presentation

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Chapter 2 Probability Math 371 University of Hawaii at M anoa Summer 2011 W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 1 / 8 Outline Chapter 2 1 Examples Definition and illustrations

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Chapter 2

Probability Math 371

University of Hawai‘i at M¯ anoa

Summer 2011

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 1 / 8

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SLIDE 2

Outline

1

Chapter 2 Examples Definition and illustrations

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 2 / 8

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Examples of some important basic concepts

Example 1. bushel of apples

proportion P(A) = |A| |Ω| (2.1.1) (2.1.3) and (2.1.4).

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 3 / 8

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Examples of some important basic concepts

Example 1. bushel of apples

proportion P(A) = |A| |Ω| (2.1.1) (2.1.3) and (2.1.4).

Example 3. toss of a “perfect” die

equally likely outcomes events mutually exclusive events relative frequency limiting frequency

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 3 / 8

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Outline

1

Chapter 2 Examples Definition and illustrations

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 4 / 8

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Definition of Probability Measure

functions

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8

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Definition of Probability Measure

functions “probability” function, P; a function defined on sets. Definition of power set P(Ω) and examples.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8

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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.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8

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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), whereas the function P itself is called a probability measure. The values of P are the probabilities of various events.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 5 / 8

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Experiments, outcomes, and events

Each point ω ∈ Ω represents a possible outcome of an experiment.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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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.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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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.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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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.”

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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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?

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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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 (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 6 / 8

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Experiments, outcomes, and events

Example 4. “favorable” outcomes and poor old D’Alembert.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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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.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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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!

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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SLIDE 19

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...

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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SLIDE 20

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!

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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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}.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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SLIDE 22

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 (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 7 / 8

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Experiments, outcomes, and events

Example 5. Roll five dice.

Find the probability they all show different faces.

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8

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Experiments, outcomes, and events

Example 5. Roll five dice.

Find the probability they all show different faces. Outcomes: What is Ω and |Ω|?

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8

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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?

  • W. DeMeo (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8

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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 (williamdemeo@gmail.com)

Chapter 2: Probability math.hawaii.edu/∼williamdemeo 8 / 8