Ch 5.1 to 5.3 Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition. Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Experiment – any activity that leads to well-defined results called outcomes. Outcome – the result of a single trial of probability experiment. Sample Space, – the set of all possible outcomes of a probability experiment. Event, – a subset of sample space. Simple event, – an event with one outcome is called a simple event. Compound event – consists of two or more outcomes. Los Angeles Mission College Presented by DW

Example 1 : A die is tossed one time Example 1 : A die is tossed one time. (a) List the elements of the sample space . (b) List the elements of the event consisting of a number that is greater than 4. (a) (b) Example 2 : A coin is tossed twice. List the elements of the sample space , and list the elements of the event consisting of at least one head. Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective B : Requirements for Probabilities 1. Each probability must lie on between 0 and 1. 2. The sum of probabilities for all simple events in equals 1. If an event is impossible, the probability of the event is 0. If an event is a certainty, the probability of the event is 1. An unusual event is an event that has a low probability of occurring. Typically, an event with a probability less than 0.05 is considered as unusual. Probabilities should be expressed as reduced fractions or rounded to three decimal places. Los Angeles Mission College Presented by DW

Example 1 : A probability experiment is conducted Example 1 : A probability experiment is conducted. Which of these can be considered a probability of an outcome? (a) (b) (c) Yes No No Los Angeles Mission College Presented by DW

Example 2 : Why is the following not a probability model? Color Probability Red 0.28 Green 0.56 Yellow 0.37 Condition 1 : Condition 2 : Check : Condition 2 was not met. Los Angeles Mission College Presented by DW

Example 3 : Given : and Find : Condition 1 : Condition 2 : Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective C : Calculating Probabilities C1. Approximating Probabilities Using the Empirical Approach (Relative Frequency Approximation of Probability) The probability of event is approximately the number of times event is observed divided by the number of repetitions of the experiment. Example 1 : Suppose that you roll a die 100 times and get six 80 times. Based on these results, what is the probability that the next roll results in six? Los Angeles Mission College Presented by DW

Example 2 : During a sale at men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought a sweater that was not white. Los Angeles Mission College Presented by DW

(a) Between 30 and 39 years of age Example 3 : The age distribution of employees for this college is shown below: Age # of Employees Under 20 25 20 – 29 48 30 – 39 32 40 – 49 15 50 and over 10 If an employee is selected at random, find the probability that he or she is in the following age groups. (a) Between 30 and 39 years of age (b) Under 20 or over 49 years of age Presented by DW Los Angeles Mission College

C2. Classical Approach to Probability (Equally Likely Outcomes are required) If an experiment has equally likely outcomes and if the number of ways that an event can occur in , then the probability of , , is If is the sample space of this experiment, where is the number of outcomes in event , and is the number of outcomes in the sample space. Los Angeles Mission College Presented by DW

(a) Compute the probability of the event . Example 1 : Let the sample space be . Suppose the outcomes are equally likely. (a) Compute the probability of the event . (b) Compute the probability of the event “an odd number.” Los Angeles Mission College Presented by DW

Example 2 : Two dice are tossed Example 2 : Two dice are tossed. Find the probability that the sum of two dice is greater than 8. Los Angeles Mission College Presented by DW

Example 3 : If one card is drawn from a deck, find the probability of getting (a) a club; (b) a 4 and a club. (a) a club Los Angeles Mission College Presented by DW

(b) a 4 and a club Los Angeles Mission College Presented by DW

(b) What is the probability that Mark will finish last? Example 4 : Three equally qualified runners, Mark, Bill, and Alan, run a 100-meter sprint, and the order of finish is recorded. (a) Give a sample space . (b) What is the probability that Mark will finish last? (a) (b) Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Event A and B are disjoint (mutually exclusive) if they have no outcomes in common. Addition Rule for Disjoint Events If E and F are disjoint events, then . Los Angeles Mission College Presented by DW

Example 1: A standard deck of cards contain 52 cards Example 1: A standard deck of cards contain 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or three from a deck of cards. Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective B : General Addition Rule The General Addition Rule For any two events E and F, . Los Angeles Mission College Presented by DW

Example 1 : A standard deck of cards contain 52 cards Example 1 : A standard deck of cards contain 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or club from a deck of cards. Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective C : Complement Rule If E represents any event and Ec represents the complement of E, then e.g. The chance of raining tomorrow is 70%. What is the probability that it will not rain tomorrow? Los Angeles Mission College Presented by DW

Let event , event , and event . Example 1 : A probability experiment is conducted in which the sample space of the experiment is . Let event , event , and event . (a) List the outcome in E and F. Are E and F mutually exclusive? Since there are common elements of 5, 6, and 7 between E and F, E and F are not mutually exclusive. (b) Are F and G mutually exclusive? Explain. Yes, there is no common element between F and G. Los Angeles Mission College Presented by DW

(d) Determine using the General Addition Rule. (c) List the outcome in E or F. Find by counting the number of outcomes in E or F. (d) Determine using the General Addition Rule. Los Angeles Mission College Presented by DW

(f) Determine using Complement Rule. (e) List the outcome in EC. Find by counting the number of outcomes in EC. (f) Determine using Complement Rule. Los Angeles Mission College Presented by DW

Example 2: In a large department store, there are 2 managers, 4 department heads, 16 clerks, and 4 stock persons. If a person is selected at random, (a) find the probability that the person is a clerk or a manager; (b) find the probability that the person is not a clerk. (a) (b) Los Angeles Mission College Presented by DW

(a) Verify that this is a probability model. Example 3: The following probability show the distribution for the number of rooms in U.S. housing units. Rooms Probability One 0.005 Two 0.011 Three 0.088 Four 0.183 Five 0.230 Six 0.204 Seven 0.123 Eight or more 0.156 Source: U.S. Censor Bureau (a) Verify that this is a probability model. Is each probability outcome between 0 and 1? Yes Is ? Yes Los Angeles Mission College Presented by DW

Approximate 89.6% of housing unit has four or more rooms. (b) What is the probability that a randomly selected housing unit has four or more rooms? Interpret this probability. Approximate 89.6% of housing unit has four or more rooms. Los Angeles Mission College Presented by DW

Example 4: According the U. S Example 4: According the U.S. Censor Bureau, the probability that a randomly selected household speaks only English at home is 0.81. The probability that a randomly selected household speaks only Spanish at home is 0.12. (a) What is the probability that a randomly selected household speaks only English or only Spanish at home? (b) What is the probability that a randomly selected household speaks a language other than only English at home? (c) Can the probability that a randomly selected household speaks only Polish at home equal to 0.08? Why or why not? No, because i.e. Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective D : Contingency Table A Contingency table relates two categories of data. It is also called a two-way table which consists of a row variable and a column variable. Each box inside a table is called a cell. Example 1 : In a certain geographic region, newspapers are classified as being published daily morning, daily evening and weekly. Some have a comics section and other do not. The distribution is shown here. (CY) (M) (E) (W) (CN) 6 9 5 7 3 15 Have Comics Section Morning Evening Weekly Yes 2 3 1 No 3 4 2 Los Angeles Mission College Presented by DW

If a newspaper is selected at random, find these probabilities. (a) The newspaper is a weekly publication. (b) The newspaper is a daily morning publication or has comics . (c) The newspaper is a weekly or does not have comics . Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Two events are independent if the occurrence of event E does not affect the probability of event F. Two events are dependent if the occurrence of event E affects the probability of event F. Example 1 : Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: The battery in your cell phone is dead. F: The battery in your calculator is dead. Independent (b) E: You are late to class. F: Your car runs out of gas. Los Angeles Mission College Presented by DW Dependent

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective B : Multiplication Rule for Independent Events If E and F independent events, then . Example 1 : If 36% of college students are underweight, find the probability that if three college students are selected at random, all will be underweight. Independent case Los Angeles Mission College Presented by DW

Example 2 : If 25% of U. S. federal prison inmates are not U. S Example 2 : If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will be U.S. citizens. Independent case Los Angeles Mission College Presented by DW

Section 5.1 Probability Rule Objective A : Sample Spaces and Events Objective B : Requirements for Probabilities Objective C : Calculating Probabilities Section 5.2 The Additional Rules and Complements Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events Objective B : General Addition Rule Objective C : Complement Rule Objective D : Contingency Table Section 5.3 Independence and Multiplication Rule Objective A : Independent Events Objective B : Multiplication Rule for Independent Events Objective C : At-Least Probabilities Los Angeles Mission College Presented by DW

Objective C : At-Least Probabilities Probabilities involving the phrase “at least” typically use the Complement Rule. The phrase at least means “greater than or equal to.” For example, a person must be at least 17 years old to see an R-rated movie. Los Angeles Mission College Presented by DW

What is the opposite of at least one correct? None is correct. Example 1 : If you make random guesses for four multiple-choice test questions (each with five possible answers), what is the probability of getting at least one correct? Direct method : Indirect method : What is the opposite of at least one correct? None is correct. Los Angeles Mission College Presented by DW

Let A be the return with income of $100,000 or more being audited. Example 2 : For the fiscal year of 2007, the IRS audited 1.77% of individual tax returns with income of $100,000 or more. Suppose this percentage stays the same for the current fiscal year. (a) Would it be unusual for a return with income of $100,000 or more to be audited? Yes, 1.77% is unusually low chance of being audited. (In general, probability of less than 5% is considered to be unusual.) (b) What is the probability that two randomly selected returns with income of $100,000 or more to be audited? Let A be the return with income of $100,000 or more being audited. Los Angeles Mission College Presented by DW

(c) What is the probability that two randomly selected returns with income of $100,000 or more will NOT be audited? (d) What is the probability that at least one of the two randomly selected returns with income of $100,000 or more to be audited? Los Angeles Mission College Presented by DW