Ch5 Probability _____________ is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. If we flip a coin 100 times and compute the proportion of heads observed after each toss of the coin, what will the proportion approach? The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. Ch 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 a probability experiment. Sample space, S – the set of all possible outcomes of a probability experiment. Event, E – a subset of the sample space Simple event, ei – an event with one outcome is called a simple event. Compound event – consists of two or more outcomes. Example 1: A die is tossed one time. (a) List the elements of the sample space S. (b) List the elements of the event consisting of a number that is greater than 4. Example 2: A coin is tossed twice. List the elements of the sample space S, and list the elements of the event consisting of at least one head. 1 Objective B : Requirements for Probabilities 1. Each probability must lie on between 0 and 1. (0PE ( ) 1) 2. The sum of the probabilities for all simple events in S equals 1. (Pe (i ) 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. Example 1: A probability experiment is conducted. Which of these can be considered a probability of an outcome? (a) 2/5 (b) -0.28 (c) 1.09 Example 2: Why is the following not a probability model? Color Probability Red 0.28 Green 0.56 Yellow 0.37 Example 3: Given: 푆 = {푒1, 푒2, 푒3, 푒4} 푃(푒1) = 푃(푒2) = 0.2 and 푃(푒3) = 0.5 Find: 푃(푒4) 2 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. Frequency of E PEE( ) RelativeFrequencyof Number of Trials of Experiement 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? 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. E Example 3: The age distribution of employees for this college is shown below: Age Number 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 3 C2. Classical Approach to Probability (Equally Likely Outcomes are required) If an experiment has n equally likely outcomes and if the number of ways that an event E can occur in m , then the probability of E , PE(), is Number of ways that Em can occur PE() Number of possible outcomes n If S is the sample space of this experiment, NE() PE() NS() where NE()is the number of outcomes in event E , and NS()is the number of outcomes in the sample space. Example 1: Let the sample space be S 1,2,3,4,5,6,7,8,9,10. Suppose the outcomes are equally likely. (a) Compute the probability of the event F 5,9 . (b) Compute the probability of the event E = "an odd number." Example 2: Two dice are tossed. Find the probability that the sum of two dice is greater than 8? Example 3: If one card is drawn from a deck, find the probability of getting (a) a club; (b) a 4 and a club 4 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 S . (b) What is the probability that Mark will finish last? Ch 5.2 The Addition 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 PEFPEPF( or ) ( ) ( ) . Example 1. A standard deck of cards contains 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. Objective B : General Addition Rule The General Addition Rule For any two events and , PEFPEPFPEF( or ) ( ) ( ) ( and ). 5 Example 1 : A standard deck of cards contains 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. Objective C : Complement Rule Complement Rule If E represents any event and Ec represents the complement of , then PEPE(C ) 1 ( ) e.g. The chance of raining tomorrow is 70%. What is the probability that it will not rain tomorrow? Example 1: A probability experiment is conducted in which the sample space of the experiment is S 1,2,3,4,5,6,7,8,9,10,11,12. Let event E 2,3,5,6,7, event F 5,6,7,8 , and event G 9,11 (a) List the outcome in EF and . Are mutually exclusive? (b) Are FG and mutually exclusive? Explain. (c) List the outcome in EF or . Find PEF( or ) by counting the number of outcomes in . (d) Determine using the General Addition Rule. 6 (e) List the outcomes in EC . Find PE()C by counting the number of outcomes in . (f) Determine PE()C using the Complement Rule. 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. Example 3: The following probability shows the distribution for the number of rooms in U.S. housing units. (a) Verify that this is a probability model. (b) What is the probability that a randomly selected housing unit has four or more rooms? Interpret this probability. 7 Example 4: According to the U.S. Census 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 0.08? Why or why not? 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 the 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. Have comics Section Morning Evening Weekly Yes 2 3 1 No 3 4 2 If a newspaper is selected at random, find these probabilities. (a) The newspaper is a weekly publication. 8 (b) The newspaper is a daily morning publication or has comics. (c) The newspaper is published weekly or does not have comics. E Ch 5.3 Independence and the Multiplication Rule Objective A : Independent Events Two events are independent if the occurrence of event does not affect the probability of event F . Two events are dependent if the occurrence of event affects the probability of event . Example 1: Determine whether the events E and F are independent or dependent. Justify your answer. (a) : The battery in your cell phone is dead. : The battery in your calculator is dead. (b) : You are late to class. : Your car runs out of gas. 9 Objective B : Multiplication Rule for Independent Events If and F are independent events, then PEFPEPF( and ) ( ) ( ) 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. E 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. 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.