Question
MATH399 Applied Managerial Statistics
Week 3 Assignment Probability Terminology and Notation
QuestionA deck of cards contains RED cards numbered 1,2,3,4,5 and BLUE cards numbered 1,2,3,4. Let Rbe the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.
Drawing the Red 5 is an example of which of the following events? Select all correct answers.   
E′
R OR E
B AND O
B AND E
 
R AND O
O′
QuestionA university offers finance courses numbered 1,2,3,4,5 and accounting courses numbered 1,2,3,4,5,6. Let F be the event of selecting a finance course, A the event of selecting an accounting course, E the event of selecting an even numbered course, and O the event of selecting an odd course.
Selecting the accounting course number 3 is an example of which of the following events? Select all correct answers.
A AND O             
F OR E  
F AND O
F OR O
E′
A′
QuestionThe average emergency room cares for 300 patients each day. There are 20 nurses on duty at the emergency room each day, and they share the patient load equally.
Suppose one patient is chosen at random.
Identify the numbers of each of the following:
QuestionA computer randomly generates numbers 1 through 100 for a lottery game. Every lottery ticket has 7numbers on it.
Identify the correct experiment, trial, and outcome below:
The experiment is the computer randomly generating a number.
The experiment is the computer randomly generating a number less than 10.
A trial is one number generated.
The trial is identifying the number generated.
An outcome is the number 2 being generated.
The outcome is the number being randomly generated.
QuestionA bowl of candy contains 6 chocolate candies and 4 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal, rounded to the hundredths place.
QuestionA bowl of candy contains 5 chocolate candies and 3 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal rounded to three decimal places.
QuestionA motor company is manufacturing pickup trucks, sedans, minivans, SUVs, ATV, and motorcycles. The dots in the Venn diagram below show the type of each vehicle. A vehicle is selected at random.
•             Let A be the event of selecting a four-wheeled vehicle.
•             Let B be the event of selecting a pickup truck or a motorcycle.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA restaurant is offering chicken specials cards numbered 1,2,3,4,5,6 and fish specials numbered 1,2,3,4,5.
Let:
•             C be the event of selecting a chicken special,
•             F be the event of selecting a fish special,
•             E be the event of selecting an even numbered special, and
•             O be the event of selecting an odd special.
Selecting the fish special number 3 is an example of which of the following events?
F′
C OR E
C AND O
E′
O′
F AND O
QuestionTwo fair dice are rolled, one blue, (abbreviated B) and one red, (abbreviated R). Each die has one of the numbers {1,2,3,4,5,6} on each of its faces. The dots in the Venn diagram below show the number and the color of the dice.
•             Let A be the event of rolling an even number on either of the dice.
•             Let B be the event of rolling a number greater than 4 on either of the dice.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA standard six-sided die shows a number, 1, 2, 3, 4, 5, or 6, on each of its sides. You roll the die once. Let E be the event of rolling the die and it showing an even number on top and L be the event of rolling a number less than 4.
Rolling a 3 is an outcome of which of the following events? Select all correct answers.
E OR L
E′ AND L
E AND L
E′ OR L′
L′
E AND L′
Probability Notation: AND, OR, and NOT
Using “OR,” “AND,” and “NOT” to Describe Events
In probability and statistics, we often are interested in combinations of events. Below, we give some of the combinations that are of particular interest. You will eventually learn formulas to compute the probabilities of these combinations of events. For now, we are interested in describing events with this notation.
“OR” Events
An outcome is in the event A OR B if the outcome is in A, or is in B, or is in both A and B. This can be represented by a Venn diagram, as shown below:
A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
Here, we see A represented by the blue circle, and notice that A={1,3,4,5,10} . B is represented by the orange circle, and B={3,5,10,12}. Then A OR B={1,3,4,5,10,12}, all of the numbers found in event A OR event B. Notice that 3, 5 and 10 are each listed once, even though they appear in both A and B.
In addition to the OR notation, you are also likely to see the notation ∪ used. You may have seen this notation used to represent the union of two sets. Similarly here, we are taking the union of two events.  A OR B and  A∪B mean the same thing.
“AND” Events
An outcome is in the event A AND B if the outcome is in both A and B at the same time. Consider again the Venn diagram:
A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
 We see that A={1,3,4,5,10} and B={3,5,10,12}. Then A AND B={3,5,10}, the numbers found in both events.
The notation you will often see for AND is ∩. You may have seen this notation used to represent the intersection of two sets. Similarly here, we are looking at the intersection of the two events A and B .
A ANDB and A∩B mean the same thing.
“NOT” Events
The complement of event A is denoted A′ (read “A prime”). A′ consists of all outcomes that are NOT in A. Refer one more time to the diagram:
A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
We see that A={1,3,4,5,10}. Here A′={12}, all of the elements of the sample space, U, that are not in A.  Sometimes you may see A′ written as ∼A.
Example 1
Question
A deck of cards for a board game contains RED cards numbered 1,2,3,4 and BLUE cards numbered 1,2,3, as shown below.
A deck of cards with blue cards numbered 1, 2, and 3 and with red cards numbered 1, 2, 3, 4.
The cards are shuffled and placed face down on the game board. Let:
•             R be the event of drawing a red card;
•             B be the event of drawing a blue card;
•             and E be the event of drawing an even numbered card.
Use “And,” “Or,” and “Not” to describe the following events:
(a) Drawing a Blue 2.
(b) Drawing a Red 1.
(c) Drawing an even card.
Example 2
Each dot outside the Venn diagram below represents a student in a particular major. The red dots represent health sciences majors (abbreviated HSM), and the blue dots represent business majors (abbreviated BM).
Let:
•             A: the event that students are currently enrolled in a writing course.
•             B: the event that students are currently enrolled in a statistics course.
Arrange the dots on the Venn diagram so that the following situation is represented (you might not use all of the dots):
Four friends are enrolled in writing and statistics courses. Of those four friends, two are health sciences majors and two are business majors.
•             Both business majors are enrolled in statistics courses, B={BM1,BM2};
•             Both health science majors are only enrolled in writing courses, A={HSM1,HSM2};
•             One of the business majors is enrolled in both a statistics course and a writing course. A∩B={BM1}
QuestionA deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.
Drawing the Blue 1 is an example of which of the following events? Select all correct answers.
R AND E
R AND O
B AND O
R′
B AND E
B′
QuestionA CEO decides to award her employees that have met their objectives this year. Those employees that have met their objectives have the chance to win vacation days. They can win either Mondays (abbreviated M) or Tuesdays (abbreviated T). They can also win up to two days. The Venn Diagrams below show the different combinations that an employee can win.
•             Let A be the event of winning two of the same day.
•             Let B be the event of winning a Monday
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA pharmaceutical company is testing their new brand of Vitamins: A, B, and C. Each type of vitamin has been labeled with a number {1,2,3} for anonymous testing so that three versions of each vitamin type are tested. The dots in the Venn diagram below show the number and the type of vitamin being tested. A researcher selects a vitamin at random.
•             Let A be the event of selecting vitamin A.
•             Let B be the event of selecting a vitamin labeled with an even number.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionThree fair coins are flipped at the same time. Each coin has the two possible outcomes: heads or tails. There are 8 possible outcomes for the three coins being flipped: {HHH,TTT,HHT,HTT,THH,TTH,HTH,THT}.
Let A be the event that the second coin flipped shows a head.
Identify the numbers of each of the following:
P(A)=$$12, is the probability that the second coin flipped shows heads.
QuestionDoctors at a research hospital are conducting a study to see if flu patients received a flu vaccination in the past year. To collect their data, they ask each person treated for the flu at their hospital whether they received the vaccination.
Identify the correct experiment, trial, and outcome below:
The experiment is recording the number of patients who say that they have received the flu vaccine.
The experiment is asking whether a flu patient received the vaccine.
A trial is asking one patient if they have been vaccinated.
The trial is identifying recording the number of patients who say that they have received the flu vaccine.
An outcome is no, the person was not vaccinated.
An outcome is asking one specific person whether they have received the flu vaccination.
QuestionA bowl of candy contains 7 chocolate candies and 5 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal, rounded to the hundredths place.
QuestionA bowl of candy contains 8 chocolate candies and 6 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal, rounded to the hundredths place.
QuestionA deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.
Which of the following events include the outcome of drawing a blue 1?
R AND O
B OR E
R OR E
B AND E
B AND O
R AND E
QuestionA mathematics professor is organizing her classroom into groups for the final project.  Each student will either be working on a graphing (G) project or writing a paper (P).  Also, each student will be working on an economics (E), finance (F), sociology (S), or criminal justice (C) problem.  The dots in the Venn diagram below show the different scenarios.
•             Let A be the event of a student working on a graphing project.
•             Let B be the event of a student writing a paper.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA deck of cards contains RED cards numbered 1,2,3,4 and BLUE cards numbered 1,2,3, as shown below.
 A deck of cards with blue cards numbered 1, 2, and 3 and with red cards numbered 1, 2, 3, 4.
 Let R be the event of drawing a red card, B be the event of drawing a blue card, E be the event of drawing an even numbered card, and O be the event of drawing an odd numbered card.
Drawing the Red 3 is an outcome in which of the following events? Select all correct answers.
R AND O
B OR E
R′
E′
E OR R
QuestionDifferent types of advertising methods are being considered for a company’s new product: a magazine ad (M), a television ad (T), a newspaper coupon (N), a radio ad (R), a coupon mailer (C), and a social media ad (S). The coupons are both good for ten dollars off the item. The dots in the Venn diagram below show the various methods. An advertising specialist considers a method at random to review.
•             Let A be the event of selecting a method that gives a discount.
•             Let B be the event of selecting a printed method.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA track team contains sprinters numbered 1,2,3,4,5 and distance runners numbered 1,2. Let S be the event of selecting a sprinter, D the event of selecting a distance runner, E the event of selecting an even numbered runner, and O the event of selecting an odd runner.
                 Selecting the sprinter numbered 1 is one outcome of which of the following events? Select all correct answers.
D AND O
D OR E
D OR O
O′
E′
S AND E
QuestionLet G be the event that a randomly chosen employee of a restaurant is a General Manager. Let S be the event that a randomly chosen employee of a restaurant works at a seafood restaurant. Identify the answer which expresses the following with correct notation: Given that the employee is a General Manager, the probability that a randomly chosen employee of a restaurant works at a seafood restaurant.
P(S|G)
P(G|S)
P(G AND S)
P(S) AND P(G)
QuestionLet M be the event that a randomly chosen student passes a math test. Let S be the event that a randomly chosen student studies every day. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen student studies every day, given that the student passes a math test.
P(M|S)
P(M AND S)
P(S) AND P(M)
P(S|M)
QuestionThere are 52 cards in a standard deck of cards, with four of each type of card: Ace ,2,3,4,5,6,7,8,9,10,Jack ,Queen ,King .
Let event A be choosing a 7 out of a deck of cards.
Identify the numbers of each of the following. Enter the probability as a fraction:
QuestionPeople with some college education stay married to their spouses for at least twenty years at a rate of 67%. How likely is it that a randomly selected, college educated couple will be married for at least twenty years?
Very likely, the probability is close to 1.
Somewhat likely, the probability is closer to 1 than to 0.
Unlikely, the probability is close to 0.
Somewhat unlikely, the probability is closer to 0 than it is to 1.
Equally likely, the probability is 0.5.
QuestionA bowl of candy contains 7 chocolate candies and 6 lemon candies. Choosing one piece of candy at random, find the probability of choosing a chocolate candy. Write your answer as a decimal, rounded to the hundredths place.
QuestionA bag contains 7 red beads, 10 blue beads, and 5 green beads. If a single bead is picked at random, what is the probability that the bead is green?
1022
522
722
1222
1722
1522
QuestionA biologist has a number of butterfly specimens. The butterflies are of various colors and various ages. The colors are blue (abbreviated B), red (abbreviated R), or yellow (abbreviated Y). Each specimen is labeled with one the numbers {1,2,3,4,5,6} and the number represents how many months old it is. The dots in the Venn diagram below show the age and the color of the specimen. The biologist selects a specimen at random.
•             Let A be the event of selecting a yellow specimen.
•             Let B be the event of selecting a specimen that is older than 2 months old.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA lawyer has numbered the cases that he is working on. He has criminal cases numbered 1,2,3,4and civil numbered 1,2,3,4,5. Let R be the event of selecting a criminal case, C the event of selecting a civil case, E the event of selecting an even numbered case, and O the event of selecting an odd case.
Selecting the criminal case number 4 is an example of which of the following events? Select all correct answers.
C′
R AND O
C OR O
C AND O
C AND E
R AND E
QuestionTwo fair dice are rolled, one blue, (abbreviated B) and one red, (abbreviated R). Each die has one of the numbers {1,2,3,4,5,6} on each of its faces. The dots in the Venn diagram below show the number and the color of the dice.
•             Let A be the event of rolling an even number on either of the dice.
•             Let B be the event of rolling a number greater than 4 on either of the dice.
Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.
QuestionA standard six-sided die shows a number, 1, 2, 3, 4, 5, or 6, on each of its sides. You roll the die once. Let E be the event of rolling the die and it showing an even number on top and L be the event of rolling a number less than 4.
Rolling a 3 is an outcome of which of the following events? Select all correct answers.
E OR L
E′ AND L
E AND L
E′ OR L′
L′
E AND L′
Probability Notation: AND, OR, and NOT
Using “OR,” “AND,” and “NOT” to Describe Events
In probability and statistics, we often are interested in combinations of events. Below, we give some of the combinations that are of particular interest. You will eventually learn formulas to compute the probabilities of these combinations of events. For now, we are interested in describing events with this notation.
“OR” Events
An outcome is in the event A OR B if the outcome is in A, or is in B, or is in both A and B. This can be represented by a Venn diagram, as shown below:
A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
Here, we see A represented by the blue circle, and notice that A={1,3,4,5,10} . B is represented by the orange circle, and B={3,5,10,12}. Then A OR B={1,3,4,5,10,12}, all of the numbers found in event A OR event B. Notice that 3, 5 and 10 are each listed once, even though they appear in both A and B.
In addition to the OR notation, you are also likely to see the notation ∪ used. You may have seen this notation used to represent the union of two sets. Similarly here, we are taking the union of two events.  A OR B and  A∪B mean the same thing.
“AND” Events
An outcome is in the event A AND B if the outcome is in both A and B at the same time. Consider again the Venn diagram:
 A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
 We see that A={1,3,4,5,10} and B={3,5,10,12}. Then A AND B={3,5,10}, the numbers found in both events.
The notation you will often see for AND is ∩. You may have seen this notation used to represent the intersection of two sets. Similarly here, we are looking at the intersection of the two events A and B .
A ANDB and A∩B mean the same thing.
“NOT” Events
The complement of event A is denoted A′ (read “A prime”). A′ consists of all outcomes that are NOT in A. Refer one more time to the diagram:
A Venn diagram with universal set Upper U contains two intersecting circles labeled Upper A and Upper B that divide Upper U into three regions. There are values in each region as follows, where the location of the region is given first and the content is given second: Upper A only, 1, 4; Upper A and Upper B only, 3, 5, 10; Upper B only, 12.
We see that A={1,3,4,5,10}. Here A′={12}, all of the elements of the sample space, U, that are not in A.  Sometimes you may see A′ written as ∼A.
Example 1
Question
A deck of cards for a board game contains RED cards numbered 1,2,3,4 and BLUE cards numbered 1,2,3, as shown below.
A deck of cards with blue cards numbered 1, 2, and 3 and with red cards numbered 1, 2, 3, 4.
The cards are shuffled and placed face down on the game board. Let:
•             R be the event of drawing a red card;
•             B be the event of drawing a blue card;
•             and E be the event of drawing an even numbered card.
Use “And,” “Or,” and “Not” to describe the following events:
(a) Drawing a Blue 2.
(b) Drawing a Red 1.
(c) Drawing an even card.
BBR′R′∪E∪E′∪E∪E′
Example 2
Each dot outside the Venn diagram below represents a student in a particular major. The red dots represent health sciences majors (abbreviated HSM), and the blue dots represent business majors (abbreviated BM).
Let:
•             A: the event that students are currently enrolled in a writing course.
•             B: the event that students are currently enrolled in a statistics course.
Arrange the dots on the Venn diagram so that the following situation is represented (you might not use all of the dots):
QuestionA deck of cards contains RED cards numbered 1,2,3 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.
Drawing the Red 1 is an example of which of the following events? Select all correct answers.
R′
B′           
O′
R OR E
B OR O
B OR E
QuestionA deck of cards contains RED cards numbered 1,2,3,4,5 and BLUE cards numbered 1,2,3,4. Let Rbe the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.
Drawing the Red 5 is an example of which of the following events? Select all correct answers.
E′
R OR E
B AND O
B AND E
R AND O
O′
QuestionKaty is deciding which charity to donate to.  She is going to donate fifty dollars to each charity chosen, and she can donate to a women’s shelter (abbreviated B), a charity for rescue animals (abbreviated R), and a children’s foundation (C). The dots in the Venn diagram below show the combinations that she can donate to.
•             Let X be the event of donating exactly one hundred dollars.
•             Let Y be the event of donating to the charity for rescue animals.
Move the dots on the Venn diagram to place the dots in the correct event, X, Y,or X AND Y. Note that you might not use all of the dots.
QuestionWhich of the following pairs of events are independent?
You roll a die twice.
Event A is getting an even number on the first roll.
Event B is getting a 4 on the second roll.
You roll a die twice.
Event A is getting a 6 on the first roll.
Event B is getting a total of more than 7.
You flip a coin and roll a die.
Event A is getting heads on the coin.
Event B is getting a 3 or more on the die.
You roll a die and flip a coin.
Event A is getting heads with the coin and getting 5 on the die.
Event B is getting 3 or more on the die.
Basic Probability Definitions
Key Terms
•             Independent events: events that have no influence on each other; two events A and B are said to be independent if the fact that one event has occurred does not affect the probability that the other event will occur
•             Dependent events: events that influence the occurrence of the other; if whether or not one event occurs does affect the probability that the other event will occur
•             Mutually exclusive: events which are impossible to both occur or that have no outcomes in common
Mutually exclusive events are also commonly referred to as Disjoint events
QuestionBeth is performing an experiment to check if a die is fair. She rolls the die 5 times and records the sequence of numbers she gets.
Which of these is an outcome of this experiment? Select all correct answers.
Rolling a die
Rolling a die five times
Rolling the sequence 1,1,2,1,6
Rolling five 4’s
Rolling the sequence 1,1,2
Basic Probability Definitions
Key Terms
•             Event: some subset of the possible outcomes of an experiment, can be described by listing the outcomes or described in words
•             Outcome: any of the possible results of an experiment in probability
Outcome is also known as Outcome space
•             Trial: one repetition or instance of a repeated experiment
QuestionJacqueline will spin a fair spinner with the numbers 0, 1, 2, 3, and 4 a total of 3 times. If Event A = spinner lands on numbers all greater than 2 and Event B = total sum of 9, which of the following best describes events A and B?
independent
dependent
mutually exclusive
complement
QuestionA fair spinner contains the numbers 1, 2, 3, 4, and 5. For an experiment, the spinner will be spun 5times. If Event A = the spinner lands on numbers all less than 3, what is an outcome of Event A?
a total sum less than 10
spinner lands on 1, 3, 1, 2, 1
a total sum of 11
spinner lands on 1, 2, 1, 2, 2
QuestionA university offers finance courses numbered 1,2,3,4,5 and accounting courses numbered 1,2,3,4,5,6. Let F be the event of selecting a finance course, A the event of selecting an accounting course, E the event of selecting an even numbered course, and O the event of selecting an odd course.
Selecting the accounting course number 3 is an example of which of the following events? Select all correct answers.
A AND O
F OR E
F AND O
F OR O
E′
A′
QuestionA deck of cards contains RED cards numbered 1,2,3,4,5,6 and BLUE cards numbered 1,2,3,4,5.
Let:
•             R be the event of drawing a red card,
•             B be the event of drawing a blue card,
•             E be the event of drawing an even numbered card, and
•             O be the event of drawing an odd card.
Drawing the Blue 3 is an example of which of the following events?
B′
R OR E
R AND O
E′
O′
B AND O
QuestionLet C be the event that a randomly chosen cancer patient has received chemotherapy. Let E be the event that a randomly chosen cancer patient has received elective surgery. Identify the answer which expresses the following with correct notation: Of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery.
P(C|E)
P(E|C)
P(E) AND P(C)
P(C AND E)
QuestionWhat is a particular result of an experiment?
event
sample space
trial
outcome
QuestionTrial best fits which of the following descriptions?
a particular result of an experiment
a subset of the set of all outcomes of an experiment
one repetition or instance of an experiment
the set of all possible outcomes of an experiment
QuestionLet B be the event that a randomly chosen person has low blood pressure. Let E be the event that a randomly chosen person exercises regularly. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person exercises regularly, given that the person has low blood pressure.
P(E|B)
P(B|E)
P(E) AND P(B)
P(B AND E)