Chapter 2. Probability. Math 371. University of Hawai i at Mānoa. Summer 2011

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1 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

2 Outline 1 Chapter 2 Examples Definition and illustrations W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 2 / 8

3 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

4 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

5 Outline 1 Chapter 2 Examples Definition and illustrations W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 4 / 8

6 Definition of Probability Measure functions W. DeMeo Chapter 2: Probability math.hawaii.edu/ williamdemeo 5 / 8

7 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

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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 Example 4. favorable outcomes and poor old D Alembert. W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 7 / 8

17 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

18 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

19 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

20 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

21 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

22 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

23 Example 5. Roll five dice. Find the probability they all show different faces. W. DeMeo Chapter 2: Probability math.hawaii.edu/ williamdemeo 8 / 8

24 Example 5. Roll five dice. Find the probability they all show different faces. Outcomes: What is Ω and Ω? W. DeMeo Chapter 2: Probability math.hawaii.edu/ williamdemeo 8 / 8

25 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

26 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

Lecture notes for probability. Math 124

Lecture notes for probability. Math 124 Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result