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35 Chapter 5 Probability Section 5.1 Probability Rules Objectives: Apply the rules of probabilities Compute and interpret probabilities using the empirical method Compute and interpret probabilities using the Classical method 5-2 Probability is a measure of the likelihood of a random phenomenon or chance behavior. The long-term proportion with which a certain outcome is observed is the probability of that outcome. 5-3 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. In probability, an experiment is any process that can be repeated in which the results are uncertain. Sample Space: the collection of all possible outcomes of an experiment. It is denoted S Event: a collection of outcomes from a probability experiment (may consist of one outcome or more than one outcome). We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the probability experiment of having two children. (a) Identify the outcomes of the probability experiment. (b) Determine the sample space. (c) Define the event E = “have one boy”. (a) (b) (c) 5-6 A probability model lists the possible outcomes of a probability experiment and each outcome’s probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities. 5-8 If an event is impossible, the probability of the event is …... If an event is a certainty, the probability of the event is …... An unusual event is an event that has a low probability of occurring. 5-9 EXAMPLE A Probability Model In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. Rule1: Rule2: Color Probability Brown 0.12 Yellow 0.15 Red 0.12 Blue 0.23 Orange 0.23 Green 0.15 5-11 5-12 EXAMPLE Building a Probability Model Pass the PigsTM is a Milton-Bradley game in which pigs are used as dice. Points are earned based on the way the pig lands. There are six possible outcomes when one pig is tossed. A class of 52 students rolled pigs 3,939 times. The number of times each outcome occurred is recorded in the table at right. (Source: http://www.members.tripod.com/~passpigs/prob.html) Outcome Frequency Side with no dot 1344 Side with dot 1294 Razorback 767 Trotter 365 Snouter 137 Leaning Jowler 32 (a) Use the results of the experiment to build a probability model for the way the pig lands. (b) Estimate the probability that a thrown pig lands on the “side with dot”. (c) Would it be unusual to throw a “Leaning Jowler”? 5-13 (a) Outcome Side with no dot Side with dot Razorback Trotter Snouter Leaning Jowler (b) (c) Probability 5-15 The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. 5-16 EXAMPLE Computing Probabilities Using the Classical Method Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected. (a) What is the probability that it is yellow? (b) What is the probability that it is blue? (c) Comment on the likelihood of the candy being yellow versus blue. (a) There are a total of ….. candies, so N(S) =…... (b) P(blue) = (c) 5-17 Section 5.2 Probability Rules 5-18 5-19 Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. 5-20 We often draw pictures of events using Venn diagrams. These pictures represent events as circles enclosed in a rectangle. The rectangle represents the sample space, and each circle represents an event. For example, suppose we randomly select a chip from a bag where each chip in the bag is labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let E represent the event “choose a number less than or equal to 2,” and let F represent the event “choose a number greater than or equal to 8.” These events are disjoint as shown in the figure. 5-21 5-22 EXAMPLE The Addition Rule for Disjoint Events The probability model to the right shows the distribution of the number of rooms in housing units in the United States. (a) Verify that this is a probability model. Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 Source: American Community Survey, U.S. Census Bureau 5-23 Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 (b) What is the probability a randomly selected housing unit has two or three rooms? P(two or three)= 5-24 Number of Rooms in Housing Unit Probability One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079 Nine or more 0.080 (c) What is the probability a randomly selected housing unit has one or two or three rooms? P(one or two or three) 5-25 5-26 5-27 EXAMPLE Illustrating the General Addition Rule Suppose that a pair of dice are thrown. Let E = “the first die is a two” and let F = “the sum of the dice is less than or equal to 5”. Find P(E or F) using the General Addition Rule. 5-28 N (E) N (S ) 6 36 1 6 P( E ) N (F ) N (S ) 10 36 5 18 P( F ) N ( E and F ) N (S ) 3 36 1 12 P( E and F ) P( E or F) = P(E) +P(F) – P(E and F) 5-29 5-30 Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted EC, is all outcomes in the sample space S that are not outcomes in the event E. 5-31 Complement Rule If E represents any event and EC represents the complement of E, then P(EC) = 1 – P(E) 5-32 EXAMPLE Illustrating the Complement Rule According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? P(do not own a dog) = 1 – P(own a dog) 5-33 EXAMPLE Computing Probabilities Using Complements The data to the right represent the travel time to work for residents of Hartford County, CT. (a) What is the probability a randomly selected resident has a travel time of 90 or more minutes? There are a total of 24,358 + 39,112 + … + 4,895 = 393,186 residents in Hartford County, CT. The probability a randomly selected resident will have a commute time of “90 or more minutes” is Source: United States Census Bureau 5-34 (b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes. P(less than 90 minutes) = 1 – P(90 minutes or more) 5-35 Section 5.3 Independence and Multiplication Rule 5-36 5-37 Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F. © 2010 Pearson Prentice Hall. All rights reserved 5-38 EXAMPLE Independent or Not? (a) Suppose you draw a card from a standard 52-card deck of cards and then roll a die. The events “draw a heart” and “roll an even number” are independent because the results of choosing a card do not impact the results of the die toss. (b) Suppose two 40-year old women who live in the United States are randomly selected. The events “woman 1 survives the year” and “woman 2 survives the year” are independent. (c) Suppose two 40-year old women live in the same apartment complex. The events “woman 1 survives the year” and “woman 2 survives the year” are dependent. 5-39 5-40 5-41 EXAMPLE Computing Probabilities of Independent Events The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that two randomly selected 60 year old females will survive the year? The survival of the first female is independent of the survival of the second female. We also have that P(survive) = 0.99186. P First survives and second survives P First survives P Second survives (0.99186)(0.99186) 0.9838 5-42 EXAMPLE Computing Probabilities of Independent Events A manufacturer of exercise equipment knows that 10% of their products are defective. They also know that only 30% of their customers will actually use the equipment in the first year after it is purchased. If there is a one-year warranty on the equipment, what proportion of the customers will actually make a valid warranty claim? We assume that the defectiveness of the equipment is independent of the use of the equipment. So, P defective and used P defective P used (0.10)(0.30) 0.03 5-43 5-44 EXAMPLE Illustrating the Multiplication Principle for Independent Events The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that four randomly selected 60 year old females will survive the year? P(all four survive) = P (1st survives and 2nd survives and 3rd survives and 4th survives) 5-45 EXAMPLE Computing “at least” Probabilities The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that at least one of 500 randomly selected 60 year old females will die during the course of the year? P(at least one dies) = 1 – P(none die) 5-47 5-48 Section 5.4 Conditional Probability and the General Multiplication Rule 5-49 5-50 Conditional Probability The notation P(F | E) is read “the probability of event F given event E”. It is the probability of an event F given the occurrence of the event E. 5-51 EXAMPLE An Introduction to Conditional Probability Suppose that a single six-sided die is rolled. What is the probability that the die comes up 4? Now suppose that the die is rolled a second time, but we are told the outcome will be an even number. What is the probability that the die comes up 4? First roll: Second roll: S = {1, 2, 3, 4, 5, 6} S = {2, 4, 6} P(S ) 1 6 P(S ) 1 3 5-52 5-53 EXAMPLE Conditional Probabilities on Belief about God and Region of the Country A survey was conducted by the Gallup Organization conducted May 8 – 11, 2008 in which 1,017 adult Americans were asked, “Which of the following statements comes closest to your belief about God – you believe in God, you don’t believe in God, but you do believe in a universal spirit or higher power, or you don’t believe in either?” The results of the survey, by region of the country, are given in the table below. Believe in God Believe in universal spirit Don’t believe in either East 204 36 15 Midwest 212 29 13 South 219 26 9 West 152 76 26 (a) What is the probability that a randomly selected adult American who lives in the East believes in God? (b) What is the probability that a randomly selected adult American who believes in God lives in the East? 5-54 Believe in God Believe in universal spirit Don’t believe in either East 204 36 15 Midwest 212 29 13 South 219 26 9 West 152 76 26 (a) What is the probability that a randomly selected adult American who lives in the East believes in God? P believes in God lives in the east N believe in God and live in the east N live in the east (b) What is the probability that a randomly selected adult American who believes in God lives in the East? P lives in the east believes in God N believe in God and live in the east N believes in God 5-55 EXAMPLE Murder Victims In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 1998, 16.6% of all murder victims were 20 – 24 year old males. What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20 - 24 years old? P male 20 24 P male and 20 24 P 20 24 = 5-56 Section 5.5 Counting Techniques If a task consists of a sequence of choices in which there are P selections for the first choice, Q selections for the second choice, R selections for the third choice, and so on, then the task of making these selections can be done in P·Q·R· … different ways. 5-58 Example: How many mix and match outfits can you make from 5 pair of jeans and 3 shirts? Example: How many license plates can be formed containing 3 letters followed by 3 numbers? (repeats allowed) ___ ____ ____ ____ ____ ____ (no repeats allowed) ___ ___ ___ ____ ___ _____ 5-59 Factorials: n! = n(n –1)(n – 2)(n – 3) · · · 3 · 2 · 1 Example: 5! = (MATH, PRB, #4) 10! = Factorial Rule – N items can be arranged in N! different ways. Example: You are car shopping and there are 8 different Nissan dealerships in the Dallas Metroplex. If you want to visit all 8 dealerships, how many different routes are possible? Example: How many different ways can 7 different video games be arranged on a shelf? A permutation is an ordered arrangement in which r objects are chosen from n distincts objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects. 5-61 Number of Permutation of distinct Objects taken r at a time The number of arrangements of r objects chosen from n objects, in which 1. The n objects are distinct, 2. Repetition of objects is not allowed, and 3. Order is important, Is given by the formula Evaluate 7P3 ; 6P6 nPr = n! / (n-r)! (MATH, PRB, #2) Example: How many ways can a President, Vice-President, Secretary, and Treasurer be chosen from 10 committee members? Example: How many different 5-letter words can be formed from the letters in the word “decagon”? Example: Lisa has 15 framed pictures. She wants to pick 5 of them to hang on the wall. How many different ways can she hang 5 pictures on the wall? A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol nCr represents the number of combinations of n distinct objects taken r at a time, where r < n. 5- Determine the value of 9C3. (MATH, PRB, #3) 5-66 The United States Senate consists of 100 members. In how many ways can 4 members be randomly selected to attend a luncheon at the White House? Example: How many ways can you pick 6 committee members from a group of 20 people? Example: How many different random samples of size 5 can be obtained from a population whose size is 30? 5-67 Computing Probabilities using Permutations and Combinations: Example: The Texas Lottery: Pick 6 numbers from 1 – 54. (Order does not matter) How many different number combinations are there? What is the probability of winning the Texas Lottery? Example: Outside a home, there is a keypad that can be used to open the garage if the correct four-digit code is entered. The numbers on the keypad are 0 – 9. How many codes are possible? What is the probability that a burglar would enter the correct code on the first try? Finding Probabilities Using “Categories” Example: A committee consists of 8 Democrats and 2 Republicans. Suppose you randomly select 4 people from the committee. Find the following probabilities: a) P (3 Democrats, 1 Republican) b) P (all 4 are Democrats) Example: Suppose you have received a shipment of 100 TVs. You don’t know it, but 6 of the TVs are defective. To determine if you are going to accept the shipment, you randomly select 5 TV and test them. If all 5 TVs work, you accept the shipment, otherwise, you reject it. What is the probability that you accept the shipment? Example: You pick 4 cards from a standard deck of cards. What is the probability of getting all Aces?