ST 201D: Activity 2 Assignment

ST 201D: Activity 2ProbabilityDue Monday February 6th at 11:59pm PST40 PointsOverview:The goal of this activity is to practice and solve problems using probability concepts, tools and laws. There aretwo parts of this activity, the first part is intended for you to practice solving probability problems by followingalong with the examples and watching the complimentary example videos. The second part is a gradedassessment. Concepts are covered in the notes, examples and videos from week 4 and chapters 12 and 13 in thetextbook. Please review these materials prior to attempting this activity.Learning Outcomes:After this activity you should be able to: Define terms: probability, sample space, event, union, intersection, compliment, independence, disjoint(mutually exclusive), random variable, finite probability model and continuous probability model. Know and apply the general addition rule and general multiplication rule. Describe and calculate expressions for probabilities. Describe and calculate conditional and independent probability problems using probability theorems. Construct and interpret Venn diagrams. Describe and calculate for random variables with finite probability distributions. Describe and calculate for random variables with continuous probability models: uniform Extract probabilities/proportions from a table of data.Materials Needed: TI-84 Calculator and Scanner to upload ActivityPart I: Probability Practice Practice example problems and watch activity 2 example videos.Watch extra probability videos within lessons. Part II: Graded Portion. Upload your completed activity by the due date in the Activity 2 link on Canvascontained in the Week 4 Module. Please only upload the graded portion. Upload as a .pdf or Word Doc.You may neatly hand write or type your answers.Feel free to discuss activity with class members however your final solutions should be your own!Duplicate activities will be considered as cheating and students involved will be reported in violation ofstudent conduct. 1 Part I: ProbabilitySome helpful hints for finding probabilities:1) Define your sample space. Make a list of all possible outcomes.2) Define the event in which you are interested in finding the probability of. Make a list or count events thatapply.3) Find a proportion. # of events that apply# of possible outcomes = # of times the event ofinterest occurstotal # of events insample space = Probability of event occurring. Example 1: A population of students were sampled and asked what type of music they listen to Rap music orCountry music or both. The probability that a randomly chosen student listens to Rap is 0.60. The probability thata randomly chosen student listens to Country is 0.40. The probability that a student listens to both rap and countrymusic is 0.25Notice how the “AND” (also callthe intersection) makes up part Rap 0.35 0.25 0.25 Country0.15 Rap AND Country of the probability for both Rapand Country music. All theprobabilities should add to 1 inthe Venn diagram. a. What is the probability that the student listens to either Rap music or Country music or both?Use the general addition rule.P(Rap music OR Country music) = P(Rap music)+ P(Country music)- P(Rap music AND Country music)= 0.60+0.40-0.25= 0.75There is a 75% chance that a student listen to either Rap music or Country music or both. b. What is the probability that the student likes only Country music?P(only Country music) = P(Country music)- P(Rap music AND Country music) = 0.60-0.25= 0.15.We can also see this in the Venn Diagram.There is a 15% chance that a student listens to only Country music and not Rap music. c. What is the probability that the student does not listen to either genre?P(no Rap music and no Country music) = 1- P(Rap music OR Country music) = 1 –[P(Rap music)+ P(Country music)P(Rap music AND Country music)] =1- [0.60+0.40-0.25=]= 0.25 or look at the “left over” proportion in the venndiagram.There is a 25% chance that a student will not listen to either genre.d. What is the probability that student listens to Rap given they listen to country?P(Rap|Country) = ( )() 0.25 =0.40=0.625. The probability a student listen to rap given the listen to country is then 0.625.e. Is listening to Rap music independent of listening to country music?The events are independent if and only if P(Rap and Country) = P(Rap)×P(country)… 0.60 × 0.40 = 0.24 ? 0.25.The condition does not hold so these events are not independent.2 Example 2: Twenty randomly selected individuals were asked whether or not they had a twitter account (Yes orNo). Their age was also recorded as either 24 years old or younger or 25 years old or older . The followingtable gives their responses.IndividualAge categoryTwitter If a person from this group was selected at random what isthe likelihood they would:125 years old or olderYesa) have a twitter account?225 years old or olderYes# ? 12() == 20 = 60%325 years old or olderYes # 425 years old or olderYesb) be 25 years old or older?525 years old or olderYes# 25 11625 years old or olderNo(25 +) === 55% # 20725 years old or olderNo825 years old or olderNoc) be 25 years old or older and have a twitter account?925 years old or olderNo(25 + )1025 years old or olderNo# 25 + ? 5=== 25%1125 years old or olderNo # 201224 years old or youngerYesd) have a twitter account given they were 25 years old or1324 years old or youngerYesolder?1424 years old or youngerYes# 25 + ? (|25+)=1524 years old or youngerYes# 25 51624 years old or youngerYes== 45.5%1724 years old or youngerYes11e) have a twitter account given they were 24 years old or1824 years old or youngerYesyounger?1924 years old or youngerNo# 24 ? ? 2024 years old or youngerNo(|24 ?) =# 24 7= = 77.8%9The venn diagram of this scenario. Example 3: The following represents the distribution of the number of work days missed due to illnessduring the flu season at a large company.yi012345p(yi) 0.25 0.42 0.21 0.08 0.03 0.01 a. What is the probability that a randomly chosen employee has missed fewer than 2 days of work due to illnessduring the flu season? The probability of missing less than two days is simple the probability of an employeemissing 0 or 1 day. Which is 0.25 +0.42 = 0.67.b. What is the probability that a randomly chosen employee has missed 5 days of work n they have missed 3 orP(5 days AND 3 or more days)0.01more days? P(5 days |3 or more days) == 0.12 = 0.0833P(3 or more days)3 Part II: ST 201 Activity 2Scan and Submit only pages from this section.40 PointsName_______________________________________________Q1. (3 points) Consider each of the scenarios:a. Can -0.41 be the probability of some event? Why or why not?b. Can 1.29 be the probability of some event? Why or why not?c. Can 0.86 be the probability of some event? Why or why not? Q2. (2 points) Suppose today’s weather forecast on weather.com says there is a 30% probability of rain today.What is the compliment of the event “rain today”? What is the probability of that compliment? 4 Use this information to answer questions Q3-Q7.The following is the roster for a small upper-division biology class at a university. The course is offered to bothbiology and non-biology majors. Each row represents the major, class standing, and course grade.StudentCameronTamaraJennyRachelLanSamCarlosJacobLamarHeatherDanielMarcusCarrieSeanTinaRobertTaraJeremy MajorBiologyBiologyBiologyBiologyOtherBiologyBiologyBiologyBiologyBiologyOtherOtherBiologyOtherOtherBiologyOtherBiology Class StandingJuniorJuniorJuniorJuniorSeniorSeniorSeniorSophomoreSophomoreJuniorSeniorSeniorJuniorJuniorSeniorSeniorSeniorSophomore GradeAAAAABBBBBBBCCCDDF Q3. (6 points) Of the students in the class the chance a randomly selected student is:Round all probabilities to three decimal places.a. a Biology major is: P(Biology Major) = b. a senior is: P(Senior) = c. a Biology major AND a Senior is: P(Biology AND Senior) is: = d. Explain step by step as if you giving a solution to someone who does not know anything aboutprobability how you came up with your answer for part c. 5 Q4. (4 points) Complete the given Venn diagram by filling in probabilities of each section. Hints: Allpro
babilities in the diagram should add to one. Start with the intersections and work outward. The whole circlefor Biology should add to what you have in part Q3a). Q5. (2 points) In your venn diagram what event does the probability that is outside your circles represent? Q6. (3 points) Given we randomly select a Senior, what is the chance they are a Biology major? That is,what is the conditional probability that a randomly selected student is a Biology major GIVEN they are an Seniorstudent? P(Biology Major | Senior) = Q7. (3 points) Create a finite probability model for the variable Grade in the small Biology course. Roundto three decimal places.GradeProbability A B C D F 6 Q8. Students at a local university have the option of taking freshman seminars during their first year incollege. A survey of the freshmen revealed the following:At the university freshman are divided into three majors: 35% are social science majors, 40% are humanitiesmajors, and 25% are physical science majors. Among the social science majors, 50% chose to take a freshmanseminar; among the humanities majors, 75% chose to take a freshman seminar; and among the physical sciencemajors, it was 30%.a. (3 points) Draw a tree diagram depicting the scenario. b. (2 points) Given a student is a Humanities major what is the likelihood they elected to not attendthe seminar? P(No|Humanities) = c. (3 points) What is the probability a randomly selected freshman is Social Science major andthey elected to take the seminar? P(Social Science and Yes) = 7 Q9. Uniform Distributions. Sarah has just arrived at her job as a checker at a grocery store. The previouschecker states that he only had one customer in the last 15 minutes. Sarah’s shift is pretty slow so she starts towonder, “When did this customer arrive in the last 15 minutes? Did the customer arrive right at the beginningof 15 minutes (0), in the middle of the 15 minute time period or right at the end (15)?” She realizes that thetime of arrival for the customer is a random variable X that has a uniform distribution from 0-15 minutes andthat the likelihood they arrived at any time during the last 15 minutes is equally probable.a. (3 points) Draw the distribution for X from 0 to 15. Make sure to label you axes and give theheight of your density “curve”. b. (3 points) What is the probability the customer arrived within the first 3 minutes of the 15minute time interval? Draw a picture. P(X<3)= c. (3 points) What is the probability the customer arrived within the last 5 minutes of the 15minute time interval? Draw a picture. P(X>10)= 8