Unit 3 Study Guide PDF

Name: _______________________________________ RS1: Unit 3 Study Guide Probability: Experimental, Theoretical, and Simulations Probability, Topic 11, Simulations Date Knowledge Check Topic(s) New Topic(s) Check Your...

Name: _______________________________________ RS1: Unit 3 Study Guide Probability: Experimental, Theoretical, and Simulations Probability, Topic 11, Simulations Date Knowledge Check Topic(s) New Topic(s) Check Your Understanding Mon. Schoolwide Catch Up Day Oct. 21 Tues./Wed. Probability Vocabulary In general, you should be Oct. 22/23 and Diagrams finishing all the material in the Study Guide for each Thurs./Fri. Probability Vocabulary and topic. Then you should 1st Quarter Exam Review check your answers with Oct. 24/25 Diagrams Mon. the key posted on 1st Quarter Exam Review Schoology. If you have Oct. 28 Tues./Wed. 1st Quarter Exam and Red any questions, ask in class Oct. 29/30 Bonus Quiz or sign up for an 8th period for extra review time. Thurs. Extra Blue Day! Oct. 31 Most topics also have Conditional Probability practice in the Google Doc Wed. Mathematical Probability for the unit. The problems Nov. 6 Rules in the Google Doc also have solutions in the key. Thurs./Fri. MAP Growth for FCPS Nov. 7/8 Final Project Introduction If you need more practice, Mon. there are Extra Practice Veterans’ Day Holiday! Nov. 11 problems for each topic. Mutually Exclusive Most are from the Tues./Weds. Conditional Probability and textbook, but some are in Events/Dependent and Nov. 12/13 Mathematical Rules their own Google Doc and Independent Events many in this unit in particular are included in Thurs./Fri. Mutually Exclusive and Counting Techniques and the Study Guide. Nov. 14/15 Independent/Dependent Theoretical Probability When you get to the end of the unit, complete the Mon. Review Sheet in the Study Topic 11 Nov. 18 Guide. The answers are at the end of the problems in the Study Guide. Tues./Weds. Counting Techniques and Simulations Nov. 19/20 Theoretical Probability Thurs./Fri. Topic 11 and Simulations Review Unit 3 Nov. 21/22 Mon./Tues. Unit 3 Test Nov. 25/26 Page 1 Page 2 Probability Vocabulary and Diagrams Tossing a Die Toss a die 20 times and record the results. Number Total Ratio Decimal Percent 1 2 3 4 5 6 Questions: 1. Find the probabilities using your data. a. P(1) b. P(even) c. P(1 or 3) d. P(not 4) e. P(prime) f. P(greater than 2) 2. Are your probabilities the same as your classmates? Check with students near you. 3. How could you find better approximations to the actual probabilities? You were able to answer the questions above because of what you learned in middle school and common sense. Let’s add some formal vocabulary to the phenomenon of probability. Page 3 The Vocabulary of Probability Probabilities can be empirical or theoretical. The empirical method was just illustrated in your “tossing a die” activity. The ratios are experimental or empirical probabilities. The probability is the observed relative frequency with which an event occurs. The theoretical probability for finding the probability of an event uses sample spaces. For example, when you toss a die there are six possible outcomes. If the die is fair, they are all equally likely to occur. The set S of all possible outcomes of an experiment is called the sample space. In this example, the sample space is {1, 2, 3, 4, 5, 6}. Any subset of a sample space is called an event, E. In this example, the event of “tossing a prime number” is the set {2, 3, 5}. In a sample space with equally likely outcomes, the n( E ) probability of an event of a sample space is defined as P( E ) = or the probability of an event is n( S ) the number of outcomes in the event divided by the number of outcomes in the sample space. So, the theoretical probability of tossing a prime number is ½. Probability measures how likely it is for an event to occur. The probability of an impossible event is 0. The probability of a certain event is 1. All events have probabilities between 0 and 1, inclusive. Consider an experiment that has one or more possible outcomes. Suppose you toss two dice. The event we wish to consider is the sum of the numbers shown on the dice. Tables help us organize sample spaces and outcomes. It is the first of three organizational tools that we will use. Note in this example, there are 36 outcomes in our sample space. 1 2 3 4 5 6 1. Complete the 6 by 6 array that shows the sum on each roll. 1 How many different sums are possible? 2 4 Are the sums equally likely? 3 4 9 2. Here are examples of 3 different events related to the dice 5 toss. E = {the sum of the dice is a prime number} 6 F = {the two dice show the same number} G = {the sum is odd} Use the table you completed to find the probability of each event listed above. a. P(E) = b. P(F) = c. P(G) = d. What is the probability that E does not occur? This is called the complement of E and is denoted EC. Another application that uses tables to organize outcomes in a sample space can be found in Biology. Page 4 Trees and Venn Diagrams A second organizational tool in probability is a tree diagram. It is most helpful when there is a sequence of events. Suppose a student tosses a coin three times. First, determine the sequence of events. In this case, it is the 1st toss, 2nd toss, and then the 3rd toss. Use the tree to show the possible outcomes. Because the branch segments are equally likely, we can determine the probabilities by simply counting. Example: a. What is the probability that there are exactly 2 heads? b. What is the probability of at least 2 heads? c. What is the probability of no heads? d. What is the probability of at most 2 tails? A third method of displaying information about sample spaces is the use of Venn diagrams. Venn diagrams are most helpful when a probability question provides information about the events in the form of the probability of the various events, the number of items per set, or the percentage of each set. A Venn Diagram is made up of two or more circles. It is often used in probability to show relationships between events. S Rectangle ‘S’ represents a sample space of possible outcomes. It’s A B also referred to as the universe. Circles ‘A’ and ‘B’ each represent specific events in the sample space S. There are three terms to remember when dealing with probability and Venn diagrams. The English word not involves complements and is designated AC or A’. In probabilistic terms, interest centers on the probability that an event will fail to occur. The English word and involves set intersection. Probabilistic problems with this word include the probability of the events overlapping or the events being in common. An intersection is designated with . The English word or involves set union. Probabilistic problems with this word include the probability of both events occurring meaning an event is in one set, the other set, or both sets. An intersection is designated with . Page 5 Practice: 1. There are three bags containing chips. The first has chips labeled 1, 2, 3, and 4. The second has chips labeled 1, 2, and 3. And, the third has chips labeled 1 and 2. You choose one chip from bag 1, then one from bag 2, and then one from bag 3. Draw a tree diagram illustrating this situation. Then answer the questions that follow. Diagram: a. What is the probability of just one 2 in the outcome? b. What is the probability of at least one 3 in the outcome? c. What is the probability that the sum of the outcomes is greater than 7? d. What is the probability that the sum of the outcomes is prime? 2. A box contains one each of $1, $2, $5, $10 and $20 bills. Two bills are chosen at random WITHOUT REPLACEMENT. Use a tree diagram to find the sample space. Then answer the questions that follow. Diagram: a. What is the probability of choosing a bill worth at least $2? b. What is the probability of choosing the $1 and $2 bill? c. What is the probability of having a total of at least $10? d. What is the probability of having a total of at most $20? Page 6 3. Jon has asked his girlfriend to make all the decisions for their date on her birthday. She will pick a restaurant and an activity for the date. Jon will choose a gift for her. The meal choices include tacos, noodles, and pizza. The activities she can choose from are Putt-Putt, bowling, and movies. Jon will buy her either candy or flowers. How many outcomes are there for these three decisions? Draw a tree diagram to illustrate the choices. The costs are listed in the table below. Answer the Dinner Cost Activity Cost Gift Cost following questions based on your tree diagram and Tacos $20 Putt-Putt $14 Candy $10 chart. Noodles $25 Bowling $10 Flowers $20 Pizza $15 Movies $20 a. If all the possible outcomes are equally likely, what is the probability that the date will cost at least $50? b. What is the probability that the date costs exactly $60? c. What is the probability that the date costs under $40? 4. We will begin by looking at a Venn diagram with values filled in. Suppose 100 students are asked what sports they participate in. Use the diagram to answer the questions below. Suppose that the event of playing basketball is represented with B, the event of participating in track is represented with T and the event of swimming is represented with S. a. How many total students are represented in the diagram? b. What is the probability that the students participate in sports? c. How many people participate in all three sports? So, what is P(T  S  B) ? d. How many people participate in basketball? In track? So, what is P ( B  T ) ? P(B T ) C e. How many people do not participate in basketball or track? So, what is ? f. Find P (T  S ). Find P(T  S )C. Page 7 5. Draw a Venn diagram illustrating the following information. Then answer the questions that follow. One hundred ninth graders were asked if they liked math, science, or tech. Everyone answered that they liked at least one. 56 like math 18 like math and science 43 like science 10 like science and tech 35 like tech 12 like math and tech 6 like all three subjects. Diagram: a. How many people like math only? b. How many students like science only? c. If one person is chosen at random, find P( S  M ). d. If one person is chosen at random, find P ( S  T ). e. If one person is chosen at random, find the probability that the person will NOT like science. Express this in mathematical notation. f. If one person is chosen at random, what is the probability that the person will like math but NOT tech. Express this in mathematical notation. Page 8 Rather than using actual counts in the Venn Diagram, it is possible to report probabilities. Consider the following example. Example: In the parking lot of a large mall 64% of cars are foreign F B made, 12% are the color blue, and 7.7% are blue and foreign made cars. The Venn diagram representing this information is shown at the right. What is the probability that a randomly selected car was:.563.077.043 a. A foreign car or a blue car?.317 P(F  B) =0.563 + 0.077 + 0.043 = 0.683 b. Not a foreign car and a blue car? P(FC ∩ B) = 0.043 0.077 c. A foreign car if it was blue? = 0.6417 0.077+0.043 0.317 d. Not blue if it was not a foreign car? 1−.64 = 0.8806 Likewise, a tree diagram, when labeled, has the probabilities needed for multiplying listed along the branch representing the required probability. Example: Suppose Kathy is flying from San Francisco to Washington, D. C., with a connection in Chicago. The probability that her first flight leaves on time is 0.15. If the flight is on time, the probability that her luggage will make the connecting flight in Chicago is 0.95, but if the first flight is delayed, the probability that the luggage will make it is only 0.65. The tree diagram representing this event is pictured at the right with the probabilities labeled along the branches. a. What is the probability that Kathy is late and her luggage does not arrive? b. What is the probability that Kathy’s luggage arrives in Washington, D. C. with her? Solutions: a. P(Kathy is late and luggage does not arrive) = 0.85(.35) = 0.297 or 0.298 b. P(Luggage arriving in D. C.) = P(on time and luggage makes connection) + P(not on time and luggage makes connection) = (0.15)(0.95) + (0.85)(0.65) = 0.695 Page 9 6. Common sources of caffeine are coffee, tea, and soda. Suppose that 55% of adults drink coffee 25% of adults drink tea 45% of adults drink soda and also that 15% drink both coffee and tea 5% drink all three beverages 25% drink both coffee and soda 5% drink only tea a. Draw a Venn diagram marked with this information. b. Use the Venn diagram you created to determine what percent of adults drink only soda. c. Use the Venn diagram you created to determine what percent of adults drink none of these beverages. 7. Suppose that Alice, Bob, and Carol are running for president of the Math Club. Alice has a 0.45 probability of being elected, while Bob has 0.35 probability and Carol has a 0.2 probability of being elected. Sue and Ted are the candidates for vice president. If Alice becomes president, the probability is 0.7 that she will choose Sue as her vice president. If Bob becomes president, the probability that he will choose Sue is 0.4, while if Carol becomes president, the probability is 0.6 that she will choose Sue. a. Make a tree diagram that represents the given information. b. What is the probability of the Bob-Sue combination? c. What is the probability that Sue will become Vice President? d. What is the probability that at least one man is an officer? Page 10 8. The Adams’ restaurant specializes in beef, chicken, and seafood. Of its customers, 32% order beef and 41% order chicken. Of those who order beef, 73% also order dessert. Of those who order chicken, 62% also order dessert. If 65% of the customers order dessert, what is the probability that someone who orders seafood will also order dessert? Empirical Probability vs. Theoretical Probability At the beginning of the unit, you rolled a die 20 times and based the probabilities that you calculated on the results of those rolls. The first probability that you calculated was P(1). If the die is fair, each outcome is equally likely so the P(1) should be 1/6. This is the theoretical probability. Did you achieve a probability of 1/6? Did your classmates? What if you combined your results with those of your classmates? Would the probability be 1/6? Comparison of Probability and Statistics (Elementary Statistics by Johnson and Kuby, 2012). Probability and statistics are two separate but related fields of mathematics. It has been said that “probability is the vehicle of statistics.” That is, if it were not for the laws of probability, the theory of statistics would not be possible. Let’s illustrate the relationship and difference between these two branches of mathematics by looking at two boxes. We know that the probability box contains five blue, five red, and five white poker chips. Probability tried to answer questions such as “If one chip is randomly drawn from this box, what is the chance that it will be blue?” On the other hand, in the statistics box we don’t know what the combination of chips is. We draw a sample and, based on the findings in the sample, make conjectures about what we believe to be in the box. Note the difference: Probability asks you about the chance that something specific, like drawing a blue chip, will happen when you know the possibilities (that is, you know the population). Statistics, on the other hand, asks you to draw a sample, describe the sample (descriptive statistics), and then make inferences about the population based on the information found in the sample (inferential statistics). Page 11 Before the Knowledge Check on Probability Vocabulary and Diagrams, you should be able to: Check When Skill/Concept You Understand Understand the difference between empirical and theoretical probability Given experimental results, calculate empirical probabilities of various outcomes Given a sample space, or a list of equally likely outcomes, calculate theoretical probabilities Construct a tree diagram and use it to calculate probabilities of various outcomes Construct a Venn diagram and use it to calculate various probabilities Correctly use notation such as the complement of an event (AC or A’), union of events (  ) and intersection of events (  ) Personal notes on Probability Vocabulary and Diagrams: Page 12 Extra Practice: A Biology Application of Probability Biologists use a Punnett Square to predict the gene combinations that are possible for an offspring when the genes of the parents are known. Each parent organism has two (homologous) chromosomes, each with a copy of a gene, or allele, for a particular trait. For example, a parent might have the genotype Bb for dimples. A capital B represents the dominant trait, which is having dimples. A lowercase b represents the recessive trait, which is not having dimples. a. Both parents contribute one allele to their offspring. For example, if both parents have the genotype Bb, an offspring could inherit the genotype of BB with each parent contributing one dominant B allele. What are the other possible combinations? b. This information can be displayed in a Punnett Square. Each side of the square B b represents the genotype of one parent. The Punnett Square for the offspring of the parents who both have the genotype Bb is shown at the right. If the dominant allele is B BB Bb present it will be the trait that appears. What is the probability that this offspring will b bB bb not have dimples? c. Create a Punnett Square to show the possible gene combinations for the offspring if one parent has the genotype Bb and the other has the genotype BB. d. What is the probability that the offspring of parents with Bb and BB will have dimples? e. You can show more complicated crosses when you consider two or more genes that are independent of each other. For example, pea pods can either be round (R) or wrinkled (r), yellow (Y) or green (y). What are the possible combinations of shape and color? RY Ry rY ry f. Complete the Punnett Square to show the RY possible gene Ry combinations for the pea pods: rY ry g. What is the probability that a pea pod with both parents RrYy will be round and yellow? In other words, what is the probability that there is an R and a Y present? h. What is the probability that a pea pod with both parents RrYy will be wrinkled and yellow? i. What is the probability that a pea pod with both parents RrYy will be wrinkled and green? Page 13  Extra Practice: Trees and Venn Diagrams 1. Because of the harsh winter weather in recent years, homeowners turned to supplementary methods of conserving energy—adding insulation to the home, closing off part of the home, or making use of alternate sources of heat such as wood stoves and fireplaces. In one small town it is estimated that 30% of all homeowners added insulation 40% closed off part of their homes 35% made use of alternate heat sources 12% added insulation and made use of alternate heat sources 15% added insulation and closed off part of their homes 13% closed off part of their home and made use of alternate heat sources 5% utilized all three measures. a. Draw a Venn diagram illustrating this situation. b. If a homeowner is selected at random, find the probability that the homeowner i. utilized exactly 1 energy-conserving measure. ii. utilized exactly two of three measures iii. utilized none of the measures iv. added insulation but did not close off part of the home. Page 14 2. Ramon has applied to both Princeton and Stanford. He thinks the probability that Princeton will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2. a. Make a Venn diagram with the probabilities given marked. b. What is the probability that neither university admits Ramon? c. What is the probability that he gets into Stanford but not Princeton? 3. In 2011, 12% of Americans over 40 owned an electronic reader (other than their phones) of some sort. Suppose that 43% of people with e-readers read at least 3 books last year, while among people without an e-reader, only 11% read 3 or more books during the year. a. Make a tree diagram illustrating this scenario including probabilities along the branches. b. Find the probability that a randomly selected American over 40 has read 3 or more books in the last year. c. Find the probability that a randomly selected American over 40 years hasn’t read 3 books. Page 15 Conditional Probability Many probabilities that we encounter in day-to-day life are the results of conditions that are already in place. So, a conditional probability is the probability that an event will occur, GIVEN that another event has occurred. The mathematical definition of conditional probability is the probability of event two (E2) happening given that event one (E1) has happened. This is the notation that is used in most conditional probability problems: P(E2|E1). The expression P(E2|E1) asks us to find the probability that event E2 occurs given event E1 has occurred. (The vertical line stands for the words “given that”.) Consider the following example. Suppose you select a tile from those shown at the right. You want to know the probability that the tile is shaded given that it is a circle. You can represent this as P(shaded|circle). Of the 5 tiles that are circles, 2 are shaded. 2 So, P(shaded|circle) =. 5 You read this as “the probability of shaded tile, given a circle”. Because you are only considering tiles that are circles, you have reduced the size of the sample space. It is common to organize information in tables in order to find conditional probabilities. Using Two-Way Tables Example: A utility company asked 50 of its customers whether they pay their bills online or by mail. What is the probability that a customer pays the bill online, given that the customer is male? The condition that the person selected is male limits the sample space. There are 12 + 8 = 20 male customers. Of those 20 customers, 12 pay online. 12 P(online|male) = = 0.6 20 Example: The table shows students by gender at two- and four-year colleges, and graduate schools, in 2005. You pick a student at random. What is P(female|graduate school)? The condition that the person selected is at graduate school limits the sample space. There are 1349 + 1954 = 3303 thousand students at graduate schools. Of those 3303 thousand students, 1954 are female. 1954 P(female|graduate school) =  0.592 3303 Page 16 Practice: 1. Use the table to find each probability. a. P(junior) b. P(female) c. P(senior and male) d. P(junior | female) e. P(male | senior) 2. Use the table at the right to find the indicated probabilities. Assume that an employee is randomly selected when finding the probabilities. a. P(less than high school education) b. P(earns over $30,000 and less than high school education) c. P(earns over $30,000 | has only high school education) d. P(has high school education or less | earns over $30,000) 3. Use the table below to find each probability. The table gives information about students at one school. Assume that a student is randomly selected when finding the probabilities. Favorite Leisure Activities Sports Hiking Reading Phoning Shopping Other Female 39 48 85 62 71 29 Male 67 58 76 54 68 39 a. P(sports | female) b. P(female | sports) c. P(reading | male) d. P(male | reading) e. P(hiking | female) f. P(hiking | male) Page 17 Mathematical Probability Rules We will now put formal mathematical rules to what we have been doing. 1. Complement Rule: In words, the Complement Rule says probability of A complement = one – probability of A. Symbolically we say P(AC)= 1- P(A). Example: Suppose that two dice are rolled and the sum is recorded. What is the probability that the sum is at least 3. Rather than add all of the sums from 3 to 12, inclusive, it is easier to compute the complement 1 – P(sum of 2) = 1 – 1/36 = 35/36. 2. General Addition Rule: Let A and B be two events defined in a sample space, S. In words, probability of A or B = probability of A + probability of B – probability of A and B. Symbolically we say P(A or B) = P(A) + P(B) – P(A and B). Example: Suppose that one die is rolled.. What is the probability that the value showing is prime or even? P(prime) + P(even) – P(prime and even) = 1/2 (values of 2, 3, 5) + 1/2 (values of 2, 4, 6) – 1/6 (value of 2) = 5/6. Go back to a previous example. Looking at the make up of camp counselors, suppose you were asked, what is the probability that the counselor is a junior or a female? Using the general addition rule, you compute P(junior) + P(female) – P( female and juniors) = 39/80 + 37/80 – 21/80 = 55/80. Since you were presented with the table, you could have also looked at the relevant cells and added the juniors (18 + 21) and the females you had not yet counted (16) and arrived at the same answer. Normally you do not have the option of computing P(A or B) in two ways as we do in this example. You will need to find P(A or B) by starting with P(A) and P(B) but you need the third piece of information, P(A and B). 3. General Multiplication Rule: Let A and B be two events, P(A and B) = P(A) x P(B, knowing A) or P(A and B) = P(A) x P(B|A). Look at the chart above again. Find the probability that the counselor is a junior and a female. Using the general multiplication rule, you compute P(junior) x P(female given the person is a junior) = (39/80)(21/39) = 21/80. Since you were presented with the table, you could have found the cell of female juniors and divided by the number of total counselors. Again, you normally do not have the option of computing P(A and B) in two ways. Note that the General Multiplication Rule can be rearranged to find conditional probability: We read P(B|A) as “the probability of B given A”. We can calculate this value using the formula above or if we have the information in a table we can use that instead. As it says above, we will usually only have one option in each problem, but we want to know how to work with anything we are given. Page 18 Practice: Find each of the following probabilities. 1. In a calculus class, there are 18 juniors and 10 seniors; 6 of the seniors are female and 12 of the juniors are males. If a student is selected at random, find the following: a. P(junior or female) b. P(senior or female) c. P(junior or senior) 2. If you draw a single card from a standard deck of cards, what is the probability that it is a king or a heart? 3. If P(A) = 0.6 and P(B) = 0.3 and P(B|A) = 0.2, find the following a. P(A and B) b. P(A or B) 4. Suppose in an animal shelter, 24% of the dogs are white, 56% are brown, and the rest are black. Find the probability that a dog is a. black b. not white c. brown or white 5. In a group of 35 children, 10 have blonde hair, 14 have brown eyes, and 4 have both blonde hair and brown eyes. If a child is selected at random, find the probability that the child has blonde hair or brown eyes. 6. Yan, a college senior, interviews with Google and DropBox. The probability of receiving an offer from Google is 0.35, from DropBox is 0.48, and from both is 0.15. Find the probability of receiving an offer from either Google or DropBox, but not both. Page 19 Before the Knowledge Check on Conditional Probability and Mathematical Rules, you should be able to: Check When Skill/Concept You Understand Use the correct notation for conditional probabilities Use two-way tables to calculate conditional probabilities Use the Complement Rule to calculate probabilities Use the General Addition Rule to calculate probabilities Use the General Multiplication Rule to calculate probabilities Use the General Multiplication Rule to calculate conditional probabilities Personal notes on Conditional Probability and Mathematical Rules: Page 20 Extra Practice: Conditional Probability The following table presents the results of a study to assess whether red dye #2 causes cancer in laboratory rats (Fienberg, 1980). The three variables are dosage (high or low), presence of cancerous tumor (yes or no), and whether the rat died before the end of the study or survived to the end of the 131- week study: (Rossman and Chance) a. Suppose one of these 88 rats was selected at random. Determine the conditional probability that the rat had a cancerous tumor, given it received a high dosage of the dye. b. Determine the conditional probability that the rat had a cancerous tumor, given it received a low dosage of the dye. c. Compare the values of these two conditional probabilities, and comment on what they reveal about the question of whether a higher dosage leads to a greater chance of developing cancer. d. Determine the conditional probability that the rat had a cancerous tumor, given it died before the study was completed. e. Determine the conditional probability that the rat had a cancerous tumor, given it survived until the end of the study. Extra Practice: Mathematical Probability Rules 1. The 2011 U.S. Senate consisted of 51 Democrats, 2 Independents, and 47 Republicans. Of the 17 women Senators, 12 were Democrats and 5 were Republicans. Suppose one of these 100 senators was chosen at random. (Rossman and Chance) a. Show how to use the addition rule to calculate the probability that the selected senator was a woman or a Democrat. b. Determine the conditional probability that the senator was a Democrat, given she was a woman. How does this compare to the (unconditional) probability that the senator was a Democrat? Page 21 c. Determine the conditional probability that the senator was a Republican, given he was a man. How does this compare to the (unconditional) probability that the senator was a Republican? 2. In the parking lot of a large mall 64% of cars are foreign made, 12% are the color blue, and 7.7% are blue and foreign made cars. Use the given information to find the following probabilities: a. A foreign car or a blue car? b. Not a foreign car and a blue car? 3. A survey of couples in a city found the following probabilities: The probability that the husband is employed is 0.85. The probability that the wife is employed is 0.60. The probability that both are employed is 0.55. A couple is selected at random. Find the probability that a. at least one of them is employed. b. neither is employed. 4. A and B are events defined on a sample space, with P(A) = 0.7and P(B|A) = 0.3. Find P(A and B). 5. A and B are events defined on a sample space, with P(A) = 0.6 and P(A and B) = 0.2. Find P(B|A). 6. John lives in New York City and commutes to work daily by subway or by Uber. He takes the subway 80% of the time because it costs less, and he uses Uber the other 20% of the time. When taking the subway, he arrives at work on time 70% of the time, and when he uses Uber he is on time 90% of the time. a. What is the probability that John took the subway and is at work on time on any given day? b. What is the probability that John used Uber and is at work on time on any given day? Page 22 Mutually Exclusive Events Two events that cannot happen at the same time are mutually exclusive events or disjoint events. If A and B are mutually exclusive events, then P( A and B) = 0. If you are looking at a Venn diagram, the closed areas representing each event “do not intersect”; i.e., there are no shared elements or they do not overlap. Example: You roll a standard number cube. Are the events below mutually exclusive? Rolling a 2 and a 3. You cannot roll a 2 and 3 at the same time. These events are mutually exclusive. Rolling an even number and a multiple of 3. The even numbers on a number cube are 2, 4, and 6. The multiples of 3 on a number cube are 3 and 6. There is one possible outcome that is both an even number and a multiple of 3, namely 6. These events are not mutually exclusive. Discuss: You randomly select an integer from 1 to 100. State whether the events are mutually exclusive. Explain your reasoning. a. The integer is less than 40; the integer is greater than 40. b. The integer is odd; the integer is a multiple of 4. c. The integer is less than 50; the integer is greater than 40. There is a special addition rule for mutually exclusive events because the probability of the intersection of the two events is 0. General Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) Addition Rule for Mutually Exclusive Events: P(A or B) = P(A) + P(B). An alternate notation is P( A or B) = P( A  B) To find the probability of either event A or event B occurring, you need to determine whether events A and B are mutually exclusive. Example: At a local high school, a student can take one foreign language each semester. About 37% of the students take Spanish. About 15% of the students take French. What is the probability that a student chosen at random is taking Spanish or French? “One foreign language each semester” means a student cannot take both Spanish and French. The events are mutually exclusive. So, P(Spanish or French) = P(Spanish) + P(French) = 0.37 + 0.15 = 0.52. Page 23 Practice: 1. Determine whether the following sets of events are mutually exclusive. a. One student is selected at random from the student body: the person selected is “male”, the person selected is “older than 16 years old”. b. Two dice are thrown: the total showing is “less than 6”, the total showing is “more than 10”. c. Two dice are thrown: the number of dots showing on the dice are “both odd”, “both even”, “total 7”, “total 11”. 2. One student is selected from the student body at TJ. Define the following events: F-the student selected is a freshman, S-the student selected is a sophomore, C-the student selected is registered for Chinese. a. Are the events F and S mutually exclusive? Explain. b. Are the events F and C mutually exclusive? Explain. c. Are the events S and C mutually exclusive? Explain. d. Are the events F and S complementary? Explain. e. Are the events F and C complementary? Explain. f. Are complementary events also mutually exclusive events? Explain. g. Are mutually exclusive events also complementary events? Explain. Page 24 3. M and N are mutually exclusive events. Find P(M or N). 3 1 a. P ( M ) = , P ( N ) = 4 6 b. P(M) = 10%, P(N) = 45% 4. Exactly 62% of the students in your school are under 17 years old. In addition, 4% of the students are over 18. What is the probability that a student chosen at random is under 17 or over 18? Dependent and Independent Events When the occurrence of one event affects the occurrence of a second event, the events are dependent events. Otherwise, the events are independent events. If the probability of event A remains unchanged after we know that B has occurred, the events are independent. Algebraically, this means P(A) = P(A|B). Note that the probability of event A is the same as the probability of A given B since B does not affect A. Example: Using the definition: Is each pair of events dependent or independent? Roll a number cube. Then spin a spinner. The two events do not affect each other. They are independent. Pick one flash card and then pick another from a stack of 30 flash cards without replacing the first card. Picking the first card affects the possible outcomes on the second pick. The events are dependent. Example: Using algebra: Recall the parking lot example from page 20. In the parking lot of a large mall 64% of cars are foreign made, 12% are blue, F B and 7.7% are blue and foreign made cars. The information has been summarized in the Venn diagram at the right..563.077.043 Is being a foreign car and being blue mutually exclusive? independent?.317 Not Mutually Exclusive because P(F ∩ B) = 0.077 ≠ 0 Independent because P(F|B) = 0.6427 = 0.64 = P(F) If the occurrence of one event does not affect the occurrence of a second event, the events are independent and the Special Multiplication Rule for Independent Events can be applied. Page 25 Special Multiplication Rule for Independent Events: A and B are independent events if and only if P ( A and B ) = P( A)  P( B). An alternate notation is P ( A and B ) = P ( A  B ) Example: At a picnic there are 10 diet drinks and 5 regular drinks. There are also 8 bags of fat-free chips and 12 bags of regular chips. If you grab a drink and a bag of chips without looking, what is the probability that you get a diet drink and fat-free chips. 3b. Using the data above, what is the probability that you get a regular drink and regular chips? Example: There are 6 green and 8 yellow bananas. Sam removes 2 at random. Find the probability that both bananas will be yellow. Solution: Because the selection of one banana affects the selection of the second banana, in order to find the probability, we can multiply: P(yellow) P(yellow|yellow) = 14 ∙ 13 = 13 8 7 4 Practice: 4. Suppose you have seven CDs in a box. Four are rock, one is jazz, and two are country. a. Today you choose one CD without looking, play it, and put it back in the box. Tomorrow, you do the same thing. What is the probability that you choose a country CD both days? b. Suppose that today you choose one CD without looking, play it, but you do not return it to the box. Tomorrow, you do the same thing. What is the probability that you choose a country CD both days? Page 26  5. Suppose you have 700 CDs in a box. Four hundred are rock, 100 are jazz, and 200 are country. a. Today you choose one CD without looking, play it, and put it back in the box. Tomorrow, you do the same thing. What is the probability that you choose a country CD both days? b. Suppose that today you choose one CD without looking, play it but you do not return it to the box. Tomorrow, you do the same thing. What is the probability that you choose a country CD both days? Watch Out Don’t confuse the property of independence with the property of being mutually exclusive. These properties are very different. In fact, if two events (with nonzero probabilities) are mutually exclusive events, then they cannot be independent because P(E|F) = 0 when E and F are mutually exclusive. Many students are tempted to always multiply (unconditional) probabilities to calculate the probability that both events occur. Be careful to do this only when the events are independent or when one of the probabilities is a conditional one. Do not be too quick to assume independence. Think about the context of the random process to judge whether assuming the events are independent is reasonable (e.g., complete strangers picking a movie vs. two friends). 6. Suppose that you have five books in your backpack. Three are novels, one is a biography, and one is a poetry book. Today you grab one book out of your backpack without looking and return it later. Tomorrow you do the same thing. What is the probability that you grab a novel both days? 7. Q and R are independent events. Find P(Q and R). 1 2 a. P ( Q ) = , P ( R ) = b. P(Q) = 0.7, P(R) = 0.3 8 5 8. A box contains four red and three blue marbles. Three marbles are to be randomly selected, one at a time. a. What is the probability that all three marbles will be red if the selection is done with replacement? b. What is the probability that all three marbles will be red if the selection is done without replacement? c. Are the drawings independent in either a or b? Justify your answer. Page 27 Before the Knowledge Check on Mutually Exclusive, Independent, and Dependent Events, you should be able to: Check When Skill/Concept You Understand Determine if two events are mutually exclusive Use the Addition Rule for Mutually Exclusive Events when appropriate Use reasoning to determine if two events are dependent or independent Determine if two events are independent or dependent algebraically Use the Special Multiplication Rule for Independent Events when appropriate Calculate conditional probabilities from tree Diagrams and Venn Diagrams Personal notes on Mutually Exclusive, Independent, and Dependent Events: Page 28 Extra Practice: Mutually Exclusive 1. A number cube is tossed. Find each probability. a. P(even or 3) b. P(less than 2 or even) c. P(prime or 4) 2. The graph at the right shows the types of bicycles in a bicycle rack. Find each probability. a. A bicycle is a 1-speed. b. A bicycle is a 3-speed or a 5-speed. c. A bicycle is not a 10-speed. d. A bicycle is not a 1-, 3-, or 10-speed. Extra Practice: Dependent and Independent Events 1. A class rolled a number cube 40 times and recorded an even number 23 times. What is the empirical estimate of the probability of rolling an even number? odd number? 2. Q and R are independent events. 1 1 a. P ( Q ) = , P ( R ) =. Find P(Q and R) 4 8 2 2 b. P ( Q ) = , P ( Q and R ) =. Find P(R) 7 9 c. P(Q) = 0.4, P(R) = 0.15. Find P(Q or R) Page 29 3. Two fair number cubes are tossed. State whether the events are mutually exclusive. a. The sum is 10; the numbers are equal. b. The sum is greater than 9; one of the numbers is 2. 1 2 4. S and T are mutually exclusive events. Given P ( S ) = , P (T ) = 6 3 a. Find P(S or T). b. Find P(S and T). 5. The 2011 U.S. Senate consisted of 51 Democrats, 2 Independents, and 47 Republicans. Of the 17 women Senators, 12 were Democrats and 5 were Republicans. Suppose one of these 100 senators was chosen at random. (Rossman and Chance) Is party independent of gender among these 100 senators? More formally, is the event “selected senator is female” independent of “selected senator is a Democrat”? Explain. (You worked on this problem earlier in the packet and may want to use those calculations here.) 6. Margaret has 10 geranium plants. She knows that 5 will flower red, 3 pink and 2 white. (Give each of your answers to this problem as a fraction.) a. What is the probability that the first plant to flower is pink? b. Write the correct probability on each branch in the tree diagram below. c. What is the probability that, of the first two plants to flower, i. both are red? ii. one is red and the other is pink? iii. at least one is pink? d. What is the probability that the first three plants to flower are all white? Page 30 For Exercises 7–11, match each word or phrase in Column A to the matching item in Column B. Column A Column B _____7. dependent events A. the occurrence of one event does NOT affect the occurrence of a second event _____8. P(A and B) if A,B are independent B. two events that cannot occur at the same time events same time _____9. independent events C. P(A) + P(B) _____10. P(A or B) if A, B are mutually D. the occurrence of one event affects the occurrence exclusive events of a second event _____11. mutually exclusive events E. P(A) · P(B) For Exercises 12–14, match each item in Column A to the matching term in Column B. Column A Column B _____12. Flip a coin. Then roll a number cube. A. dependent events _____13. snow and 80°F weather B. independent events _____14. Pick a piece from a set of chess C. mutually exclusive events pieces. Then pick a second piece without replacement. 15. The tree diagram relates snowfall and school closings. Let H, L, O and C represent heavy snowfall, light snowfall, schools open, and schools closed, respectively. Find each probability. a. Find P(H and O) b. Find P(H|C) c. Find P(L|O) Page 31 Counting Techniques In the first part of this unit, we have dealt with empirical probabilities where the counting has already been completed. In the second part of the unit we will deal with theoretical probabilities. With theoretical probability, you don’t actually conduct an experiment (i.e. roll a die or conduct a survey). Instead, you use your knowledge about a situation, some logical reasoning, and/or a known formula to calculate the probability of an event happening. The first thing we must do to calculate theoretical probabilities is to examine some basic counting ideas and formulas. Permutations and Combinations It is easy to count the ways you can choose items from a short list. But sometimes there are so many choices that counting the possibilities is impractical. The Fundamental Counting Principle describes the method of using multiplication to count the number of outcomes that can occur. Fundamental Counting Principle The Fundamental Counting Principle states that if one event has m possible outcomes and a second event has n possible outcomes, then there are m x n total possible outcomes for the two events together. Example: Four different fruits and 6 different vegetables give 4 6 possible fruit and vegetable combinations. 1. John has 6 computers, 7 printers and 3 scanners to choose from. How many possible computer- printer combinations can he make? A permutation is an arrangement of items in a particular order. Suppose you want to find the number of ways to order three items. There are 3 ways to choose the first item, 2 ways to choose the second, and 1 way to choose the third. By the Fundamental Counting Principle, there are 3(2)(1) = 6 permutations. Using factorial notation, you can write 3(2)(1) as 3! and read “three factorial”. For any positive integer n, n factorial is n ! = n  (n − 1)  (n − 2)   3  2 1. Note: 0! = 1 by definition. Evaluate each expression without a calculator. 11! 2. 6! 3. 9! 9! 4. 6!3! 5. 2!6! Page 32  Number of Permutations The number of permutations of n items of a set arranged r items at a time is n! n Pr = for 0  r  n ( n − r )! Example 8! 8! P3 = = = 336 8 ( 8 − 3)! 5! Evaluate each expression. 6. 7P4 = 7. 9P5 = Example: Ten students are in a race. First, second and third places will win medals. In how many ways can 10 runners finish first, second and third place (no ties allowed)? 8. In how many ways can 15 runners finish first, second, and third? a. Use the permutation formula to set up and find the answer. b. In Problem a, is the number of ways for runners to finish first, second, and third the same as the number of ways to finish eighth, ninth, and tenth? Explain. In the previous example, we found the number of ways that 10 runners can finish in first, second, and third places. Suppose instead that the three runners who finish first, second, and third in a race advance to a championship race. In this case, the order in which the first three runners cross the finish line does not matter. A selection in which order does not matter is called a combination. Page 33  Number of Combinations The number of combinations of n items of a set chosen r items at a time is n! n Cr = for 0  r  n r !( n − r )! Example 7! 7! 7  6  5  4  3  2 1 7 65 C3 = = = = = 35 3!( 7 − 3)! 3!  4! ( 3  2 1)( 4  3  2 1) ( 3  2 1) 7 Evaluate each expression. 9. 12C11 10. 12C1 11. 12C5 12. 12C7 5 C3 13. 4C4 + 4C3 + 4C2 + 4C1 + 4C0 14. 5 C2 15. What do you observe about the pairs of problems 9 and 10 and 11 and 12? Show why this occurs. 16. Thirty people apply for 10 job openings as welders. How many different groups of people can be hired? Page 34 Before the Knowledge Check on Counting Techniques, you should be able to: Check When Skill/Concept You Understand Understand the Fundamental Counting Principal Evaluate expressions with factorials Recognize the difference between permutations and combinations Use the formula for permutations and combinations to determine the number of outcomes of various situations Personal notes on Counting Techniques: Page 35 Extra Practice: Counting Techniques Evaluate each expression. 6!2! 1. 3!4! 2. 3. 6C2 8! C5 4. 6P2 5. 2 ( 7 C5 ) 6. 7 5 C2 7. How many different orders can you choose to read six of the nine books on your summer reading list? 8. How many ways are there to choose five shirts out of seven to take to camp? 9. There are five different math books and three different biology books on a shelf. a. In how many ways can they be arranged? b. In how many ways can they be arranged if the math books are grouped together and the biology books are grouped together? 10. In 1966, one type of Maryland license plate had two letters followed by four digits. How many of this type of plate were possible? Use the Fundamental Counting Principle to find the number of possible license plates. Complete the expression. 26  ___ ___ ___ _____  _____ = 11. An art gallery plans to display seven sculptures in a single row. Write a factorial expression to show how many different arrangements of the sculptures are possible, and then evaluate it. 12. In how many ways can four distinct positions for a relay race be assigned from a team of nine runners? For each situation in 13-16, determine whether to use a permutation or a combination. Then solve the problem. 13. You draw the names of 5 raffle winners from a basket of 50 names. Each person wins the same prize. How many different groups of winners could you draw? 14. How many different 5-letter codes can you make from the letters in the word cipher? 15. Nine people tryout for the nine positions on a baseball team. In how many ways could the positions be filled if Fred must be the pitcher? 16. Using the letters SARDINE, how many different arrangements have D first and a consonant second? Page 36 Finding Theoretical Probabilities The rules we learned in the first part of the unit are the same for theoretical probabilities. Summary of Basic Probability Rules 1. For any event E, 0 ≤ P(E) ≤ 1 2. Sum of all outcomes in the sample space S is 1: P(S) = 1. number of outcomes favorable to E 3. If the outcomes in S are equally likely to occur, the P(E) = total number of outcomes in S 4. Complement Rule: P(Ec) = 1 – P(E) Rules Involving More than One Event 1. General Addition Rule: P(E  F) = P(E) + P(F) – P(E ∩ F) 2. Addition Rule for disjoint events: If E and F are disjoint (mutually exclusive) events, then P(E  F) = P(E) + P(F) 3. *General Multiplication Rule: P(F∩E) = P(E)∙P(F|E) 4. Multiplication Rule for Independent Events: P(E∩F) = P(E)∙P(F) if and only if E and F are independent events. P(F  E ) 5. *Conditional Probability Formula: P(F|E) = P(E) *Note that these 2 formulas are simple rearrangements of the 3 probabilities. Now rather than determining the numerator or denominator from a table or information, you must determine the theoretical numbers by choosing if you are using permutations or combinations and using the appropriate counting rules. Example: A permutation is selected at random from the letters SEQUOIA. What is the probability that Q is in the fourth position and that it ends in a vowel? Solution: To answer this question you must compute the number of permutations meeting the requirements. And, you must compute the number of permutations in the sample space. For the numerator, you can use a schematic. There is only one way to have a Q in the fourth position and there are 5 ways of having a vowel in the last position: ___ ∙___ ∙ ___ ∙ 1 ∙ ___ ∙ ___ ∙ 5 Since there are five letters remaining, once the Q and vowel are placed, the number of permutations is 5 ∙ 4 ∙ 3 ∙ 1 ∙ 2 ∙ 1 ∙ 5 = 600 For the denominator, we must find the number of permutations in the general sample space; i.e., how many “words” can be formed with the 7 letters. Since the letters cannot be used more than once, this is a permutation, 7! = 5040. So, the probability is n(Event) 600 5 P(Event) = n(Sample Space) 5040 42 Page 37 Example: If a committee of five is selected at random from a group of 9 people (6 girls and 3 boys), what is the probability that it will have a. John and Mary as two of the 5 people? b. Exactly 3 girls and 2 boys? c. At least 3 girls? Solution: The sample space is the set of all possible 5-member committees. This is a combination: 9 C5 126. This will serve as the denominator in each case. For part a, there is only one way to choose both John and Mary and we need three additional 1 C1 1 C1 7 C3 35 people, so we are computing: 35 P (John and Mary) = 126 For part b, there are 6 C3 ways of choosing the girls and 3 C2 ways of choosing the boys. Since both calculations must be performed the total number of ways is found by multiplying: 6 C3 3 C2 60 60 So, P ( exactly 3 girls and 2 boys ) = 126 For part c, the committees could be 3 girls and 2 boys OR 4 girls and 1 boy OR five girls. In this case, we must add the total number of ways of getting the committees. n(at least 3 girls) = n(3G, 2B) + n(4G, 1B) + n(5G, 0B)= 6 C3 3 C2 6 C4 3 C1 6 C5 3 C0 20 3 15 3 6 1 111 111 So, P(at least 3 girls)= 126 Practice: 1. Many manufactured items look interchangeable. Examples are ball bearings, light bulbs, and transistors. However, an individual ball bearing may be too large or too small, and a light bulb or a transistor that looks fine may prove to be defective. The following exercises require the computation of certain probabilities based on the number of defective items and the size of the sample. Answers can be given in terms of combination notation. a. A shipment contains 50 transistors, 3 of which are defective. What is the probability that a randomly chosen transistor from this shipment works? What is the probability that it is defective? b. A sample consisting of 2 transistors is chosen from this shipment. What is the probability that both transistors work? What is the probability that both transistors are defective? c. A shipment contains 80 ball bearings, 5 of which are defective. What is the probability that a randomly selected ball bearing meets specifications? What is the probability that it is defective? Page 38 2. A sample of three ball bearings is chosen from the above shipment. a. What is the probability that all three ball bearings are acceptable? b. What is the probability that two ball bearings are acceptable and one is defective? c. What is the probability that one is acceptable and two are defective? d. What is the probability that all three ball bearings are defective? e. What is the probability that an odd number of ball bearings are defective? 3. In a class there are 6 girls and 4 boys. a. Two students are chosen at random from the class. In how many ways can this be done? b. If it is necessary for there to be one girl and one boy, how many ways are possible? 4. A shipment of toy cars contains 40 red cars and 45 blue cars. Two cars of each color are defective. A sample of three toy cars is chosen from the shipment. a. What is the probability that all three toy cars are blue? b. What is the probability that two toy cars are blue and one is red? c. What is the probability that two toy cars are acceptable and one is defective? d. What is the probability that two toy cars are red and acceptable and one is blue and defective? e. What is the probability that one toy car is an acceptable blue car, another is a defective blue car, and the other is a defective red car? Page 39 Before the Knowledge Check on Theoretical Probability, you should be able to: Check When Skill/Concept You Understand Use counting techniques (combinations and permutations) to determine theoretical probabilities of various events Apply the appropriate probability rules (i.e. the General Addition Rule) along with counting techniques to calculate theoretical probabilities Personal notes on Theoretical Probability: Page 40 Extra Practice: Finding Theoretical Probability 1. In the original version of poker known as “straight” poker, a 5-card hand is dealt from a standard deck of 52 cards. Find the probability of the given event. a. A hand will contain at least one king. b. A hand will be a “full house” (any three of one kind and a pair of another kind.) c. A hand will contain 4 aces and a king. d. A hand will contain 4 aces. 2. In how many ways can a committee of 5 people be chosen from 7 men and 3 women if it must contain at least one woman? 3. Two members of a math department are to be chosen at random to attend a convention. If the department has 9 men and 6 women members, what is the probability that 2 men attend the conference? 4. Mr. Lightner has 13 socks in his drawer, 7 blue and 6 green. He selects 6 socks at random. What is the probability that he gets a. 3 blue and 3 green b. 4 blue and 2 green c. 2 blue and 4 green or 3 blue and 3 green Page 41 Topic 11: Probability (where the book introduces probability…) Complete Activity 11-1: Random Babies in your Google Doc. Watch Out! Use the Watch Out after Activity 11-1 to answer the following questions: 1. How will we define an unlikely event? 2. How we will define a rare event? The Law of Large Numbers While a chance process is impossible to predict in the short term, if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. Above is the cumulative proportion of tosses for a fair coin. The previous example confirms that the probability of getting a head when we toss a fair coin is 0.5. Probability 0.5 means “occurs half the time in a very large number of trials.” That doesn’t mean that you are always guaranteed 50% heads and 50% tails for any number of tosses! The Law of Large Numbers, therefore, never guarantees a specific outcome when we observe a chance process—rather, it points out that there the proportion trends towards the value that we predict for the chance process. Page 42 Expected Value An example we all have experience with is a 6-sided die. Here is the probability distribution: Outcome 1 2 3 4 5 6 Probability 1/6 1/6 1/6 1/6 1/6 1/6 The outcomes are not always equally likely, so the probabilities are not always the same like they are here. But the total probability in a distribution will always be 1. To calculate the expected value of the probability distribution for a 6-sided die, you would do the following: 1 1 1 1 1 1 1 2 3 4 5 6 1 + 2 + 3 + 4 + 5 + 6 21 1 +2 +3 +4 +5 +6 = + + + + + = = = 3.5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 That means the long-run average value for rolling a 6-sided die is 3.5. It isn’t an actual outcome, you can’t get 3.5 when you roll the die, but you can see how it is directly in the middle of the six equally likely outcomes above. Practice Problems: 1. An insurance company charges $900 annually for auto policies for teenagers. The policy specifies the company will pay $1,500 for a minor accident and $8,000 for a major accident. If the probability of a teenager having a minor accident during the year is 0.15, and of having a major accident is 0.05, how much can the insurance company expect to make on 10 policies? 2. The number of hybrid cars a dealer sells weekly has the following probability distribution: Number of hybrids 0 1 2 3 4 5 Probability 0.32 0.28 0.15 0.11 0.08 0.06 The dealer purchases the cars for $21,000 and sells them for $24,000. What is the expected weekly profit from selling hybrid cars? Page 43 Complete Activity 11-2: Random Babies in your Google Doc. Watch Out! Use the Watch Outs (there are two) after Activity 11-2 and after part a of Activity 11-3 to answer the following questions: 1. An expected value is interpreted as the long-run _____________ of a numerical random process. 2. Many people fall into the trap believing that probabilities should also hold in the _______________. Remember, probability is a long-term property. Simulations When what we want to investigate is not easy to carry out (costly, dangerous, unnecessary, etc.), we can create a simulation to run the experiment. Remember the seven steps below: Steps in Creating a Simulation 1. Identify the real-world activity that is to be repeated. 2. Link the activity to one or more random numbers. 3. Describe how you will use the random number assignment to complete a full trial. 4. State the response variable. 5. Run several trials. 6. Collect and summarize the results of all the trials. 7. State your conclusion. Use these steps to describe how to simulate each scenario below. 1. Look back at Random Babies Activity (11-1) from textbook page 226. What were the first 4 simulation steps for random babies? 2. Explain how you could conduct a simulation to determine the probability of these situations: a. Guessing the correct answer on at least 7 out of 10 true/false questions. b. Choosing a yellow tulip bulb from a bin if one in six of the bulbs in the bin is yellow. Page 44 3. What is the probability of scoring 80% or better on a five-question true/false quiz if you guess at every answer? a. Design a simulation that enables you to estimate this probability. b. Run 20 trials of your simulation and record the number of times you score 80% or better. c. Calculate and interpret the experimental probability of scoring 80% if you guess every answer. Complete “More Simulation Practice” in your Google Doc. Multiple Choice Guessing Experiment In an experiment to determine whether people can do better than guessing on a random multiple-choice question, participants are asked to read and answer a given question with four answer choices. Participants do not need to have any outside knowledge of the topic the question is related to, but on average should still perform better than simply guessing which is the correct answer. Suppose that an experiment consists of thirty of these trials. As we will learn more about later, we always identify the hypotheses of an experiment at the beginning. The first is the null hypothesis, denoted by H. The null hypothesis is typically a statement 0 of "no effect" or "no difference." It states that the parameter of interest (the value we are studying; 𝑝, proportion for categorical data and µ, mean for quantitative data) is equal to a specific, hypothesized value– that a specific action had “no effect” on the value. In the context of a population proportion, H has the form: 0 𝐻0: 𝑝 = 𝑝0 where p is the population proportion of interest and 𝑝0 is replaced by the conjectured value of interest. The second hypothesis that we identify is the alternative hypothesis is denoted by Ha. It states what the researchers or pollsters suspect, or hope is true about the parameter of interest. In this case, if we believe that the students can do better than guessing on the multiple-choice question then our alternative hypothesis is: Ha: p > p0 There are other forms of the alternative hypothesis that we will study at a later time. Complete Multiple Choice Guessing Experiment in your Google Doc. In our upcoming unit, we will learn what “convincing” evidence is. Page 45 Before the Knowledge Check on Topic 11 and Simulations, you should be able to: Check When Skill/Concept You Understand State and understand the Law of Large Numbers and how this applies to probability Know the seven steps for a simulation Use a random digits table and a random number generator Assign random numbers to outcomes in a simulation to appropriately represent the probability of each outcome Complete a simulation using the seven steps Recognize and use the notation for the null and alternative hypotheses Personal notes on Topic 11 and Simulations: Page 46 Extra Practice: Review Problems True/False 1. The probability of an event that is certain is 1. 2. 0! = 0 3. 10C1 = 10! 4. (a +0)! = a! Multiple Choice 5. You choose 5 apples from a case of 24 apples. Which best represents the number of ways you can make your selection? A. 5C19 B. 24C5 C. 5P24 D. 19P5 6. Which is equivalent to 7P3? A. 28 B. 35 C.210 D. 840 7. A traveler can choose from three airlines, five hotels, and four rental car companies. How many arrangements of these services are possible? A. 12 B. 60 C. 220 D. 495 8. In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green? A. 1/3 B. ¾ C. 7/19 D. 8/21 E. 9/21 Short Answer You need not simplify the expression. You may leave it in permutation, combination and/or factorial form. 9. A survey of college students finds that 40% like country music, 30% like gospel music, and 10% like both. a. Make a Venn diagram illustrating these results. b. What percent of college students like country but not gospel? c. What percent of college students like neither country nor gospel? 10. Decide whether to use a permutation or a combination for each situation. Then solve the problem. a. An ice cream parlor offers 14 different types of ice cream. In how many different ways can you select 5 types of ice cream to sample? b. Eleven groups entered a science fair competition. In how many ways can the groups finish first, second, and third? 11. How many 3-digit numbers can we make using the digits 2, 3, 4, 5, and 6 without repetitions? Page 47 12. How many 6 letter words can we make using the letters in the word LIBERTY without repetitions? 13. In how many ways can you arrange 5 different books on a shelf? 14. In how many ways can you select a committee of 3 students out of 10 students? 15. How many triangles can you make using 6 non-collinear points on a plane? 16. A committee including 3 boys and 4 girls is to be formed from a group of 10 boys and 12 girls. How many different committees can be formed from the group? 17. In a certain country, the car number plate is formed by 4 digits from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 followed by 3 letters from the alphabet. How many number plates can be formed if neither the digits nor the letters are repeated? 18. How many seven-digit phone numbers can be made under the following conditions: (First digit cannot be 0 or 1 because you'll get the operator or long distance). The first digit is 3 and the second digit is 6. The third digit is even. The fourth digit is greater than 5. The fifth and seventh digits are odd. The sixth digit is 2. 19. Michelle, Ruth, Ana, and Susan are trying out for the varsity basketball team. In all, 21 girls are trying out, and 12 girls will make the team. a. How many different ways can the coach choose a team? b. Seven of the 21 players were on the team last year. If they all make the team, in how many ways can the coach select from the other 14 players to complete the team? c. The coach wants to divide the players into different groups. At random, each player takes a warm- up jersey from a bin with 9 red, 7 yellow, and 5 blue jerseys. If Ana chose first, what is the probability that she gets a yellow jersey? If Michelle picks next, what is the probability that she chooses a red jersey? 20. A club has 8 male and 8 female members. The club is choosing a committee of 6 members. The committee must have 3 male and 3 female members. How many different committees can be chosen? 21. On any given day there’s a 0.8 probability that I’ll be happy. If I’m happy there’s a 0.7 probability I’ll study hard for the test, but if I’m not happy there’s only a 0.2 probability. a. Construct a tree diagram and find the probabilities that I (i) study hard and (ii) don’t study hard. b. If I study hard for the test there’s a 0.6 probability that I’ll get a good grade, but if I don’t study hard there’s only a 0.2 probability. Find the probability that I get a good grade. Page 48 22. The international club at school has 105 members, many of whom speak multiple languages. The most commonly spoken languages in the club are English, Spanish and Chinese. Use the Venn diagram at the right to determine the probability of a student who a. Does not speak English b. Speaks Spanish given that they speak English c. Speaks English given that they speak Chinese d. Speaks Spanish and English but not Chinese 23. A box contains four blue and six green balls. Two balls are randomly selected from the box. Determine the probability for the following, first (i) WITH replacement, and then (ii) WITHOUT replacement. a. Both are green b. First is blue and the second is green c. A green and a blue one are obtained 24. Classify each pair of events as dependent or independent. a. Roll a number cube. Then roll it again. b. Pull a card from a deck of playing cards. Then pull a second card. 25. Q and R are independent events. Find P(Q and R) if P(Q) = 0.8 and P(R) = 0.2. 26. Given the following table of survey results of snack preferences, find the probability that a person selected at random Chips Fruit Popcorn from a group is: Males 40 30 30 a. male Females 30 20 60 b. prefers fruit c. a male who prefers fruit d. a female who prefers fruit or a male e. a male given prefers fruit f. prefers fruit given male g. Are preferring fruit and being male independent? 27. After studying the traffic patterns at this corner over a long period of time, we determine that cars go straight 60% of the time, turn right 25% of the time, and turn left 15% of the time. Now, we form a probability model by listing the elements of our sample space, together with their associated probabilities. (See Table below). Vehicle Direction Straight Right Left Probability 0.6 0.25 0.15 a. Let C be the event that a vehicle comes to the corner and then either goes straight or turns right. Find P(C). b. Two vehicles are randomly selected from the traffic study. What is the probability that neither of them goes straight? c. Two vehicles are randomly selected from the traffic study. What is the probability that at least one of them goes straight? 28. Adam is your school’s star soccer player. When he takes a shot on goal, he typically scores half of the time. Suppose that he takes six shots in a game. To estimate the probability of the number of goals Adam makes, use simulation with a number cube. One roll of a number cube represents one shot. Page 49  a. Specify what outcome of a number cube you want to represent a goal scored by Adam in one shot. b. For this problem, what represents one trial of taking six shots? c. Perform and list the results of ten trials of this simulation. d. Identify the number of goals Adam made in each of the ten trials you did in part (c). e. Based on your ten trials, what is your estimate of the probability that Adam scores three goals if he takes six shots in a game? Once you have completed the review questions above, complete the Self-Check problem at the end of each topic. The textbook page numbers and necessary calculator lists are given below. The solution to each Self-Check follows in the textbook so you can check your answers immediately. Self-Check 11-5, textbook page 236 Answers: 1. T 2. F 3. F 4. T 5. B 6. C 7. B 8. A 9. 9b. 30% 9c. 40% 10a. combination - 14 C5 = 2002 C G 10b. permutation - 11 P3 = 990.3.1.2 11. 5 P 3 = 60 12. 7 P 6 = 5040 13. 5! = 120 14. 10 C 3 = 120 15. 6 C 3 = 20 16. 10 C 3 * 12 C 4 = 59,400 17. 9 P 4 * 26 P 3 = 47,174,400 study.56 18. Use the Fundamental Counting Principle: 1∙1∙5∙4∙5∙1∙5 = 500 phone numbers.7 19. a. 21C12 = 293,930 b. 14C5 = 2002.8 don't study c. P(yellow) = 7/21 = 1/3 P(red|yellow already selected) = 9/20 =.45 happy.3.24 20. 8C3 ∙ 8C3 = 562 = 3136 study.04 21. (a) P(study hard) =.56 +.04 =.6.2.2 P (don’t study hard) = 1 -.6 =.4 not happy don't study (b) P(good grade) = P(good grade|study hard) +.8 P(good grade|don’t study hard) =.6∙.6 +.4∙.2 =.36 +.08 =.44.16 (also known as Law of Total Probability) good.36.6 grade 16 41 21 33 22. a. b. c. d..6 not good grade 105 89 26 105 study.4.24 23. a. with replacement: P(GG) = (.6)(.6) =.36 good.08 without replacement: P(GG) = (.6)(5/9) = 1/3.4.2 grade b. with replacement: P(BG) = (.4)(.6) =.24 don't study not good grade without replacement: P(BG) = (.4)(6/9) = 4/15.8 c. with replacement: P(BG) + P(GB) = (.4)(.6) + (.6)(.4) =.48.32 without replacement: P(BG) + P(GB) = (.4)(6/9) + (.6)(4/9) = 2(4/15) = 8/15 24. a. independent b. dependent 25. 0.16 26. Chips Fruit Popcorn Total Males 40 30 30 100 Females 30 20 60 110 Total 70 50 90 210 a. male 10/21 ≈.476 b. prefers fruit 5/21 ≈.238 c. a male who prefers fruit 1/7 ≈.143 d. a male or prefers fruit 12/21 = 4/7 ≈.571 e. a male given prefers fruit 3/5 f. prefers fruit given male 3/10 g. No. Explanation: P(male) ≠ P(male|prefers fruit). Alternate explanation: P(male ∩ prefers fruit) = 1/7 ≠ P(male)∙P(prefers fruit) = 10/21 ∙ 5/21 =.113 Page 50 27. a. P(C) =.60 +.25 =.85 b. We’re sampling without replacement, so we must make the assumption that 2 cars is less than 10% of the population of cars from which we’re sampling. That assumption paves the way for claiming independence for the two vehicles. Then P(not straight ∩ not straight) =.42. c. P(at least one goes straight) = 1 – P(neither goes straight) = 1– P(not straight ∩ not straight) = 1 –.42 =.84. 28. a. Answers will vary; students need to determine which three numbers on the number cube represent scoring a goal. b. Rolling the cube six times represents taking six shots on goal or 1 simulated trial. c. Answers will vary. Performing only ten trials is a function of time. Ideally, many more trials should be done. d. Answers will vary. e. Answers will vary; the probability of scoring per shot is ½. Glossary Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) Addition Rule for Mutually Exclusive (or disjoint) Events: P(A or B) = P(A) + P(B). An alternate notation is P( A or B) = P( A  B). Combination: a way of selecting several things out of a larger group, where order does not matter n! Combination Rule: C(n, r) = nCr = r !( n − r )! Complement Rule: (video 19) the probability that an event does not occur; 1 – P(A) Complementary Events: mutually exclusive events whose probabilities sum to 1 Conditional Probability: the probability that an event, B, will occur given that another event, A, has already occurred P(F  E ) Conditional Probability Rule: P(F|E) = P(E) Cumulative Relative Frequency: the cumulative proportion or the running total of frequencies Dependent Events: when the occurrence of one event affects the occurrence of a second event Empirical Estimate of Probability: (often just written Empirical Probability) an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences. Equally Likely Outcomes: outcomes that occur with the same probability; i.e., landing on heads or tails on a fair coin toss are equally likely outcomes Event: a subset of a sample space Expected Value the long-run average result of a numerical random process Experimental Estimate of Probability: the probability of an event occurring when an experiment was conducted Page 51 Factorial (n!): n! = n(n – 1)(n – 2)... 3(2)(1) Fundamental Counting Principle: f event M can occur in m ways and is followed by event N that can occur in n ways, then event M followed by event N can occur in m n ways. Independent Events: when the occurrence of one event does not affect the occurrence of a second event Intersection: the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. Multiplication Rule: P(F∩E) = P(E)∙P(F|E) Multiplication Rule for Independent Events: P(E∩F) = P(E)∙P(F) if and only if A and B are independent events. Mutually Exclusive Events: events that cannot occur at the same time; also called disjoint events Outcome: a possible result from a probability experiment Permutation: a way of selecting several things out of a larger group, where order does matter n! Permutation Rule: P(n, r) = nPr = ( n − r )! Probability: likelihood that an event will occur and is calculated by dividing the number of favorable outcomes by the total number of possible outcomes Probability Distribution a way of representing the likelihood of all the possible results of a statistical event. Random Process (Chance Experiment): any activity or situation in which there is uncertainty about which of two or more possible outcomes will result Random Variable: the numerical outcome of a random phenomenon Sample Space: the set of all possible outcomes in an experiment Simulation a way to model random events Theoretical (Exact) Probability: a ratio of the number of favorable outcomes to the number of possible outcomes Trial: the single performance of an experiment Union: everything in both sets Zero factorial: one Page 52