ST 201D: Activity 2 Probability Due Monday February 6th at 11:59pm PST 40 Points Overview: The goal of this activity is to practice and solve problems using
probability concepts, tools and laws. There are two parts of this activity, the first part is intended for you to practice solving probability problems by following

along with the examples and watching the complimentary example videos. The second part is a graded assessment. Concepts are covered in the notes, examples and videos

from week 4 and chapters 12 and 13 in the textbook. 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 Activity Part 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 Canvas contained 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 of student conduct. 1 Part I: Probability Some 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 that apply. 3) Find a

proportion. # of events that apply # of possible outcomes = # of times the event of interest occurs total # of events in sample space = Probability of event occurring.

Example 1: A population of students were sampled and asked what type of music they listen to Rap music or Country music or both. The probability that a randomly chosen

student listens to Rap is 0.60. The probability that a randomly chosen student listens to Country is 0.40. The probability that a student listens to both rap and

country music is 0.25 Notice how the “AND” (also call the intersection) makes up part Rap 0.35 0.25 0.25 Country 0.15 Rap AND Country of the probability for both Rap

and Country music. All the probabilities should add to 1 in the 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= 0.75 There 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

venn diagram. 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 or No).

Their age was also recorded as either 24 years old or younger or 25 years old or older . The following table gives their responses. Individual Age category Twitter If

a person from this group was selected at random what is the likelihood they would: 1 25 years old or older Yes a) have a twitter account? 2 25 years old or older Yes #

? 12 () = = 20 = 60% 3 25 years old or older Yes # 4 25 years old or older Yes b) be 25 years old or older? 5 25 years old or older Yes # 25 11 6 25 years old or older

No (25 +) = = = 55% # 20 7 25 years old or older No 8 25 years old or older No c) be 25 years old or older and have a twitter account? 9 25 years old or older No (25 +

) 10 25 years old or older No # 25 + ? 5 = = = 25% 11 25 years old or older No # 20 12 24 years old or younger Yes d) have a twitter account given they were 25 years

old or 13 24 years old or younger Yes older? 14 24 years old or younger Yes # 25 + ? (|25 +) = 15 24 years old or younger Yes # 25 5 16 24 years old or younger Yes = =

45.5% 17 24 years old or younger Yes 11 e) have a twitter account given they were 24 years old or 18 24 years old or younger Yes younger? 19 24 years old or younger No

# 24 ? ? 20 24 years old or younger No (|24 ?) = # 24 7 = = 77.8% 9 The venn diagram of this scenario. Example 3: The following represents the distribution of the

number of work days missed due to illness during the flu season at a large company. yi 0 1 2 3 4 5 p(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 illness during the flu season? The probability of missing less than two days is simple the

probability of an employee missing 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 or P(5 days AND 3 or more days) 0.01 more days? P(5 days |3 or more days) = = 0.12 = 0.0833 P(3 or more days) 3 Part II: ST 201 Activity 2 Scan and

Submit only pages from this section. 40 Points Name_______________________________________________ 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 both biology and non-biology majors. Each row represents the major, class standing, and course grade. Student Cameron Tamara Jenny

Rachel Lan Sam Carlos Jacob Lamar Heather Daniel Marcus Carrie Sean Tina Robert Tara Jeremy Major Biology Biology Biology Biology Other Biology Biology Biology Biology

Biology Other Other Biology Other Other Biology Other Biology Class Standing Junior Junior Junior Junior Senior Senior Senior Sophomore Sophomore Junior Senior Senior

Junior Junior Senior Senior Senior Sophomore Grade A A A A A B B B B B B B C C C D D F 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 about probability 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: All probabilities in the diagram should add to

one. Start with the intersections and work outward. The whole circle for 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 Senior student? P(Biology Major | Senior) = Q7. (3

points) Create a finite probability model for the variable Grade in the small Biology course. Round to three decimal places. Grade Probability A B C D F 6 Q8. Students

at a local university have the option of taking freshman seminars during their first year in college. A survey of the freshmen revealed the following: At the

university freshman are divided into three majors: 35% are social science majors, 40% are humanities majors, and 25% are physical science majors. Among the social

science majors, 50% chose to take a freshman seminar; among the humanities majors, 75% chose to take a freshman seminar; and among the physical science majors, 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 attend

the seminar? P(No|Humanities) = c. (3 points) What is the probability a randomly selected freshman is Social Science major and they 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 previous checker states that he only had

one customer in the last 15 minutes. Sarah’s shift is pretty slow so she starts to wonder, “When did this customer arrive in the last 15 minutes? Did the customer

arrive right at the beginning of 15 minutes (0), in the middle of the 15 minute time period or right at the end (15)?” She realizes that the time of arrival for the

customer is a random variable X that has a uniform distribution from 0-15 minutes and that 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 the height of your density “curve”. b. (3 points) What

is the probability the customer arrived within the first 3 minutes of the 15 minute 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 15 minute time interval? Draw a picture. P(X>10)= 8

ST 201D: Activity 2 Probability Due Monday February 6th at 11:59pm PST 40 Points Overview: The goal of this activity is to practice and solve problems using
probability concepts, tools and laws. There are two parts of this activity, the first part is intended for you to practice solving probability problems by following

along with the examples and watching the complimentary example videos. The second part is a graded assessment. Concepts are covered in the notes, examples and videos

from week 4 and chapters 12 and 13 in the textbook. 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 Activity Part 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 Canvas contained 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 of student conduct. 1 Part I: Probability Some 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 that apply. 3) Find a

proportion. # of events that apply # of possible outcomes = # of times the event of interest occurs total # of events in sample space = Probability of event occurring.

Example 1: A population of students were sampled and asked what type of music they listen to Rap music or Country music or both. The probability that a randomly chosen

student listens to Rap is 0.60. The probability that a randomly chosen student listens to Country is 0.40. The probability that a student listens to both rap and

country music is 0.25 Notice how the “AND” (also call the intersection) makes up part Rap 0.35 0.25 0.25 Country 0.15 Rap AND Country of the probability for both Rap

and Country music. All the probabilities should add to 1 in the 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= 0.75 There 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

venn diagram. 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 or No).

Their age was also recorded as either 24 years old or younger or 25 years old or older . The following table gives their responses. Individual Age category Twitter If

a person from this group was selected at random what is the likelihood they would: 1 25 years old or older Yes a) have a twitter account? 2 25 years old or older Yes #

? 12 () = = 20 = 60% 3 25 years old or older Yes # 4 25 years old or older Yes b) be 25 years old or older? 5 25 years old or older Yes # 25 11 6 25 years old or older

No (25 +) = = = 55% # 20 7 25 years old or older No 8 25 years old or older No c) be 25 years old or older and have a twitter account? 9 25 years old or older No (25 +

) 10 25 years old or older No # 25 + ? 5 = = = 25% 11 25 years old or older No # 20 12 24 years old or younger Yes d) have a twitter account given they were 25 years

old or 13 24 years old or younger Yes older? 14 24 years old or younger Yes # 25 + ? (|25 +) = 15 24 years old or younger Yes # 25 5 16 24 years old or younger Yes = =

45.5% 17 24 years old or younger Yes 11 e) have a twitter account given they were 24 years old or 18 24 years old or younger Yes younger? 19 24 years old or younger No

# 24 ? ? 20 24 years old or younger No (|24 ?) = # 24 7 = = 77.8% 9 The venn diagram of this scenario. Example 3: The following represents the distribution of the

number of work days missed due to illness during the flu season at a large company. yi 0 1 2 3 4 5 p(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 illness during the flu season? The probability of missing less than two days is simple the

probability of an employee missing 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 or P(5 days AND 3 or more days) 0.01 more days? P(5 days |3 or more days) = = 0.12 = 0.0833 P(3 or more days) 3 Part II: ST 201 Activity 2 Scan and

Submit only pages from this section. 40 Points Name_______________________________________________ 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 both biology and non-biology majors. Each row represents the major, class standing, and course grade. Student Cameron Tamara Jenny

Rachel Lan Sam Carlos Jacob Lamar Heather Daniel Marcus Carrie Sean Tina Robert Tara Jeremy Major Biology Biology Biology Biology Other Biology Biology Biology Biology

Biology Other Other Biology Other Other Biology Other Biology Class Standing Junior Junior Junior Junior Senior Senior Senior Sophomore Sophomore Junior Senior Senior

Junior Junior Senior Senior Senior Sophomore Grade A A A A A B B B B B B B C C C D D F 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 about probability 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: All probabilities in the diagram should add to

one. Start with the intersections and work outward. The whole circle for 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 Senior student? P(Biology Major | Senior) = Q7. (3

points) Create a finite probability model for the variable Grade in the small Biology course. Round to three decimal places. Grade Probability A B C D F 6 Q8. Students

at a local university have the option of taking freshman seminars during their first year in college. A survey of the freshmen revealed the following: At the

university freshman are divided into three majors: 35% are social science majors, 40% are humanities majors, and 25% are physical science majors. Among the social

science majors, 50% chose to take a freshman seminar; among the humanities majors, 75% chose to take a freshman seminar; and among the physical science majors, 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 attend

the seminar? P(No|Humanities) = c. (3 points) What is the probability a randomly selected freshman is Social Science major and they 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 previous checker states that he only had

one customer in the last 15 minutes. Sarah’s shift is pretty slow so she starts to wonder, “When did this customer arrive in the last 15 minutes? Did the customer

arrive right at the beginning of 15 minutes (0), in the middle of the 15 minute time period or right at the end (15)?” She realizes that the time of arrival for the

customer is a random variable X that has a uniform distribution from 0-15 minutes and that 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 the height of your density “curve”. b. (3 points) What

is the probability the customer arrived within the first 3 minutes of the 15 minute 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 15 minute time interval? Draw a picture. P(X>10)= 8