Chapter 2 Probability Math 371 University of Hawai i at Mānoa Summer 2011 W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 1 / 8
Outline 1 Chapter 2 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 Ω (2.1.3) and (2.1.4). (2.1.1) 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 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 (2.1.1) W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 3 / 8
Outline 1 Chapter 2 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
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
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
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
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
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
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 Ω (2.1.11) W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 6 / 8
Example 4. favorable outcomes and poor old D Alembert. W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 7 / 8
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
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
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
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
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
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
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
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
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
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