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
Outline Chapter 2 1 Examples Definition and illustrations W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 2 / 8
Examples of some important basic concepts Example 1. bushel of apples proportion P ( A ) = | A | ( 2 . 1 . 1 ) | Ω | (2.1.3) and (2.1.4). W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 3 / 8
Examples of some important basic concepts Example 1. bushel of apples proportion P ( A ) = | A | ( 2 . 1 . 1 ) | Ω | (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
Outline Chapter 2 1 Examples Definition and illustrations W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 4 / 8
Definition of Probability Measure functions W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 5 / 8
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
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
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
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
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
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 ω 1 ∈ Ω . W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 6 / 8
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 ω 1 ∈ Ω . If ω 1 ∈ A , then we say “the event A has occurred.” W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 6 / 8
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 ω 1 ∈ Ω . If ω 1 ∈ 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
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 ω 1 ∈ Ω . If ω 1 ∈ 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 | ( 2 . 1 . 11 ) | Ω | W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 6 / 8
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
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 7 / 8
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. Wrong! W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 7 / 8
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. Wrong! Give D’Alembert a computer... W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 7 / 8
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. Wrong! Give D’Alembert a computer... ...he could quickly estimate P ( { ω 3 } ) ≈ 1 / 2. Extra credit! W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 7 / 8
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. Wrong! Give D’Alembert a computer... ...he could quickly estimate P ( { ω 3 } ) ≈ 1 / 2. Extra credit! The problem: ω i above are not equally likely outcomes . Instead, let Ω = { ω 1 , ω 2 , ω 3 , ω 4 } = { HH , TT , HT , TH } . W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ ∼ williamdemeo 7 / 8
Experiments, outcomes, and events Example 4. “favorable” outcomes and poor old D’Alembert. Toss two identical coins. The sample space is... Ω = { ω 1 , ω 2 , ω 3 } = { two heads, two tails, a head and a tail } ...so P ( { ω 3 } ) = 1 / 3. Wrong! Give D’Alembert a computer... ...he could quickly estimate P ( { ω 3 } ) ≈ 1 / 2. Extra credit! The problem: ω i above are not equally likely outcomes . Instead, let Ω = { ω 1 , ω 2 , ω 3 , ω 4 } = { 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
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
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
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
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
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