Question
MATH399 Applied Managerial Statistics
Week 3 Assignment Independent and Mutually Exclusive Events
QuestionGiven the following information about events A and B
•             P(A)=0
•             P(A AND B)=0
•             P(B)=0.25
Are A and B mutually exclusive, independent, both, or neither?
A and B are independent because P(A AND B)=P(A)⋅P(B).
A and B are both independent and mutually exclusive.
 A and B are mutually exclusive because P(A AND B)=0.
A and B are neither independent nor mutually exclusive.
QuestionA game show releases its secrets about how it chooses contestants from its audience. They say
•             The probability of being chosen for the first round on the show is 130;
•             The probability of being chosen for the second round on the show is 215;
•             The probability of being chosen for both rounds on the show is 0.
Let event A be being chosen for the first round, and event B being chosen for this second round. Are events A and B mutually exclusive, independent, both, or neither? 
Events A and B are mutually exclusive.
 Events A and B are independent.
 Events A and B are both mutually exclusive and independent.
 Events A and B are neither independent nor mutually exclusive. 
QuestionGiven the following information about events A and B:
•             P(A)=112
•             P(A AND B)=116
•             P(A|B)=112
Are events A and B mutually exclusive, independent, both, or neither?
Events A and B are both independent and mutually exclusive because P(A|B)=P(A AND B).
A and B are neither independent nor mutually exclusive.
 A and B are mutually exclusive since P(A|B)=P(A).
 A and B are independent since P(A|B)=P(A).
QuestionSuppose that A is the event you purchase an item from an online clothing store, and B is the event you purchase the item from a nearby store. If A and B are mutually exclusive events, P(A)=0.57,and P(B)=0.17, what is P(A|B)?
QuestionThe event of eating breakfast at a diner is A and the event of watching cable is B. If these events are independent events, using P(A)=0.22, and P(B)=0.46, what is P(B|A)?
QuestionAn evenly weighted game spinner has 5 numbers on it, labeled and contained in a colored piece of the spinner:
•             Three are colored red: 1,2,3, (abbreviated R);
•             Two are colored blue: 4, and 5 (abbreviated B).
Consider the Venn diagram below. Each of the dots outside the Venn diagram represents a number that the spinner may land on. Place the dots in the appropriate event given the information below (you may not use all of the dots), then determine if the events are mutually exclusive.
•             Event A: Spinning an even number.
•             Event B: Spinning a number greater than 3.
QuestionEvent A: Rolling an odd number on a fair die.
Event B: Rolling a 4 on a fair die.
Event C: Rolling an even number on a fair die.
Given the three events, which of the following statements is true? Select all that apply.
Event A and Event B are mutually exclusive.
Event B and Event C are mutually exclusive.
Event A and Event C are not mutually exclusive.
Event A and Event C are mutually exclusive.
 
QuestionYou have a fair die, with six faces containing the numbers 1,2,3,4,5,6. Given Events A and B, are the two events mutually exclusive? Explain your answer.
Event A: Rolling a 1 or a 2
Event B: Rolling an odd number.
Yes, the events are mutually exclusive because they have no outcomes in common.
Yes, the events are mutually exclusive because P(A) is not equal to P(B).
No, the events are not mutually exclusive because they both require rolling the same die.
No, the events are not mutually exclusive because they share the common outcome of 1.
Mutually Exclusive Events
To make calculating probability easier, we want to identify any special relationships between events. For example, if we were asked to find the probability that a triathlete is both riding their bike and swimming at the same time, we should be very skeptical.
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0.
Consider the Venn diagram above. Event A has {1,2,3,4,5} , and event B has {6,7,8,9,10} , as shown. We say that A and B are mutually exclusive events because they do not share any of the same possible outcomes. And, P(A AND B)=0.
If the possible outcomes for A were {1,2,3,4,5}, and the possible outcomes for B were {5,6,7,8,9}, as shown above in the diagram, A and B would not be mutually exclusive. Notice in the diagram that A AND B={5}, because they share the possible outcome of 5. So, P(A AND B) is not equal to zero, and A and B are not mutually exclusive.
If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.
Mutually Exclusive Events
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0.
Example 1
Question
You have a fair, well-shuffled deck of 52 cards. It consists of four suits: clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 2,3,4,5,6,7,8,9,10, J (jack), Q (queen), K (king), and A (ace) of that suit.
Given Events A and B, are the two events mutually exclusive?
Event A: Choosing a J (jack).
Event B: Choosing a card in the spades suit.
Example 2
Each dot outside the Venn diagram below represents a student in a particular major. The red dots represent mathematics majors (abbreviated MM), and the blue dots represent economics majors (abbreviated EM).
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.
First, arrange the dots on the Venn diagram so that the following situation is represented (you might not use all of the dots):
•             Both mathematics majors are enrolled in statistics courses, B={MM4,MM5};
•             Two economics majors are enrolled in writing courses, A={EM1,EM2}.
•             One economics major, {EM3},  is not enrolled in a statistics or a writing course
Question:
Use your diagram to determine whether the following statement is true or false:
Five friends, two mathematics majors and three economics majors, are enrolled in writing and statistics courses, but mathematics and economics majors do not take class together. So, events A and B are mutually exclusive.
Key Terms
•             Mutually Exclusive Events are two or more events that cannot occur at the same time.
Events that are Mutually Exclusive can also be described as Disjoint events.
QuestionConsider the Venn diagram below. Each of the dots outside the circle represents a graduating college student surveyed about their post-college job search. 
•             Five of the graduates are business majors: represented by red dots and labeled: 1,2,3,4,5 (abbreviated BM);
•             Four of the graduates are social work majors: and are represented by blue dots: 1,2,3,4(abbreviated SW).
The two events represented in the Venn diagram are:
•             Event A: The student has applied to at least one post-college job.
•             Event B: The student currently is working in an internship position.
Three of the business majors BM1,BM2,BM4 and two of the social work majors SW2,SW3 say they have applied to at least one job. Business majors BM2,BM3 state that they are currently working in an internship position. Social work majors SW2,SW4 say they are also currently working in an internship position. Students BM5,SW1 say that neither of the options currently apply to them.
Place the dots in the appropriate event given the information above (you may not use all of the dots), then determine if the events are mutually exclusive.
QuestionAt a major international airport, passengers are questioned about their destination. Given Events A and B, are the two events mutually exclusive? Explain your answer.
Event A: The passengers are traveling to Paris, France.
Event B: The passengers are NOT traveling to Paris, France.
Yes, the events are mutually exclusive because P(A) is not equal to P(B).
 No, the events are not mutually exclusive because they both require asking the passengers where they are going.
Yes, the events are mutually exclusive because they have no outcomes in common.
No, the events are not mutually exclusive because they share the common outcome of Paris, France.
QuestionAn HR director numbered employees 1 through 50 for an extra vacation day contest. What is the probability that the HR director will select an employee who is not a multiple of 13?
•             Give your answer as a fraction.
QuestionPhones collected from a conferences are labeled 1 through 40. What is the probability that the conference speaker  will choose a number that is not a multiple of 6?
Question A card is drawn from a standard deck of 52 cards. Remember that a deck of cards has four suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards: Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King. Given the three events, which of the following statements is true? Select all that apply.
Event A: Drawing a clubs.
Event B: Drawing a spade.
Event C: Drawing a Queen.
 Event B and Event C are mutually exclusive.
Event A and Event B are mutually exclusive.
Event A and Event C are not mutually exclusive.
Event A and Event C are mutually exclusive.
QuestionConsider the Venn diagram below. Each of the dots outside the circle represents a customer at a restaurant who was asked about their drink preferences. 9 customers were questioned.
•             Three of the customers are men, represented by red dots and labeled: M1,M2,M3;
•             Six of the customers are women, represented by blue dots and labeled: W1,W2,W3,W4,W5,W6.
The two events represented in the Venn diagram are:
•             Event A: The customer prefers to order a soda.
•             Event B: The customer prefers to order water.
Each of the men questioned said they prefer to order water, as did W1 and W4. The women who said they prefer to order soda were W2 and W6. All of the other customers questioned said they did not want soda or water.
Place the dots in the appropriate event given the information above (you may not use all of the dots), then determine if the events are mutually exclusive.
QuestionOn an Alaskan cruise, shore excursions are offered most days. One day’s options were:
•             Kayaking to a glacier;
•             Hiking to a waterfall.
Cruise travelers can choose to participate in one excursion,or no excursions. A family on the cruise is divided on which activity to choose. Jack and Shirley want to kayak. Emma and Chris want to hike to a waterfall. Kelly wants to stay on the ship and read her book.
Arrange the family members in their activity choice for the day in the Venn diagram below. Then, use the Venn diagram to answer the question:
Are kayaking and hiking mutually exclusive events?
QuestionWhich of the following pairs of events are mutually exclusive?
Select all correct answers.
Event A: rolling a 6-sided fair number cube and getting an even number
Event B: rolling a 6-sided fair number cube and getting a number greater than 4
Event A: rolling a 6-sided fair number cube and getting 1
Event B: rolling a 6-sided fair number cube and getting an even number
Event A: drawing a card from a 52-card standard deck and getting a red face card
Event B: drawing a card from a 52-card standard deck and getting a black card
Event A: drawing a card from a 52-card standard deck and getting a face card
Event B: drawing a card from a 52-card standard deck and getting a red card
Mutually Exclusive Events
Key Terms
•             Mutually Exclusive Events are two or more events that cannot occur at the same time.
Events that are Mutually Exclusive can also be described as Disjoint events.
QuestionAn evenly weighted game spinner has 5 numbers on it, labeled and contained in a colored piece of the spinner:
•             Three are colored red: 1,2,3, (abbreviated R);
•             Two are colored blue: 4, and 5 (abbreviated B).
Consider the Venn diagram below. Each of the dots outside the Venn diagram represents a number that the spinner may land on. Place the dots in the appropriate event given the information below (you may not use all of the dots), then determine if the events are mutually exclusive.
•             Event A: Spinning an even number.
•             Event B: Spinning a number greater than 3.
QuestionEvent A: Rolling an odd number on a fair die.
Event B: Rolling a 4 on a fair die.
Event C: Rolling an even number on a fair die.
Given the three events, which of the following statements is true? Select all that apply.
Event A and Event B are mutually exclusive.
Event B and Event C are mutually exclusive.
Event A and Event C are not mutually exclusive.
Event A and Event C are mutually exclusive.
QuestionYou have a fair die, with six faces containing the numbers 1,2,3,4,5,6. Given Events A and B, are the two events mutually exclusive? Explain your answer.
Event A: Rolling a 1 or a 2
Event B: Rolling an odd number.
Yes, the events are mutually exclusive because they have no outcomes in common.
Yes, the events are mutually exclusive because P(A) is not equal to P(B).
No, the events are not mutually exclusive because they both require rolling the same die.
No, the events are not mutually exclusive because they share the common outcome of 1.
Mutually Exclusive Events
To make calculating probability easier, we want to identify any special relationships between events. For example, if we were asked to find the probability that a triathlete is both riding their bike and swimming at the same time, we should be very skeptical.
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0.
Consider the Venn diagram above. Event A has {1,2,3,4,5} , and event B has {6,7,8,9,10} , as shown. We say that A and B are mutually exclusive events because they do not share any of the same possible outcomes. And, P(A AND B)=0.
If the possible outcomes for A were {1,2,3,4,5}, and the possible outcomes for B were {5,6,7,8,9}, as shown above in the diagram, A and B would not be mutually exclusive. Notice in the diagram that A AND B={5}, because they share the possible outcome of 5. So, P(A AND B) is not equal to zero, and A and B are not mutually exclusive.
If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.
Mutually Exclusive Events
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0.
Example 1
Question
You have a fair, well-shuffled deck of 52 cards. It consists of four suits: clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 2,3,4,5,6,7,8,9,10, J (jack), Q (queen), K (king), and A (ace) of that suit.
Given Events A and B, are the two events mutually exclusive?
Event A: Choosing a J (jack).
Event B: Choosing a card in the spades suit.
Example 2
Each dot outside the Venn diagram below represents a student in a particular major. The red dots represent mathematics majors (abbreviated MM), and the blue dots represent economics majors (abbreviated EM).
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.
First, arrange the dots on the Venn diagram so that the following situation is represented (you might not use all of the dots):
•             Both mathematics majors are enrolled in statistics courses, B={MM4,MM5};
•             Two economics majors are enrolled in writing courses, A={EM1,EM2}.
•             One economics major, {EM3},  is not enrolled in a statistics or a writing course
Question:
Use your diagram to determine whether the following statement is true or false:
Five friends, two mathematics majors and three economics majors, are enrolled in writing and statistics courses, but mathematics and economics majors do not take class together. So, events A and B are mutually exclusive.
Key Terms
•             Mutually Exclusive Events are two or more events that cannot occur at the same time.
Events that are Mutually Exclusive can also be described as Disjoint events.
Question
A fair die has six sides, with a number 1,2,3,4,5 or 6 on each of its sides. In a game of dice, the following probabilities are given:
•             The probability of rolling two dice and both showing a 1 is 136;
•             The probability of rolling the first die and it showing a 1 is 16;
•             If you roll one die after another, the probability of rolling a 1 on the second die given that you’ve already rolled a 1 on the first die is 16.
Let event A be the rolling a 1 on the first die and B be rolling a 1 on the second die. Are events A and B mutually exclusive, independent, neither, or both?
Events A and B are mutually exclusive.
Events A and B are independent.
Events A and B are both mutually exclusive and independent.
Events A and B are neither mutually exclusive nor independent.
 
QuestionGiven the following information about events B and C:
•             P(B)=70%
•             P(B AND C)=0
•             P(C)=45%
Are B and C mutually exclusive, independent, both, or neither?
B and C are both mutually exclusive and independent because P(B AND C)=0
B and C are independent because P(B AND C)=0
B and C are mutually exclusive because P(B AND C)=0.
B and C are neither mutually exclusive nor independent.
QuestionGiven the following information, determine whether events B and C are independent, mutually exclusive, both, or neither.
•             P(B)=0.6
•             P(B AND C)=0
•             P(C)=0.4
•             P(B|C)=0
Independent
Mutually Exclusive
Both Independent & Mutually Exclusive
Neither
QuestionA game requires that players draw a blue card and red card to determine the number of spaces they can move on a turn. Let A represent drawing a red card, with four possibilities 1,2,3, and 4. Let B represent drawing a blue card, and notice that there are three possibilities 1,2, and 3.
A deck of cards with blue cards numbered 1, 2, and 3 and with red cards numbered 1, 2, 3, 4.
If the probability of a player drawing a red 2 on the second draw given that they drew a blue 2 on the first draw is P(R2|B2)=14, what can we conclude about events A & B? 
Events A and B are mutually exclusive since P(R2|B2)=P(R2).
Events A and B are mutually exclusive since P(R2|B2)=P(R2)=0.
Events A and B are independent since P(R2|B2)=P(R2).
Events A and B are independent since P(R2|B2)≠P(R2).
Distinguishing Between Independent and Mutually Exclusive Events
Conditional probabilities can also tell us information about whether two events are independent or mutually exclusive.  Recall that if two events are independent that one occurring does not have any effect on the other occurring. Events that are mutually exclusive share no outcomes.
Say events A and B are independent. Then, P(A|B)=P(A), no matter what is going on with the P(B). This goes the other way as well. We can determine if two events are independent if we know information about P(A|B) and P(A).
Conditional Probability and Independent Events
Two or more events are said to be independent if the event of one occurring has no effect on whether or not the other one will also occur. We look for one of the following equivalent equations to determine independence:
•             P(A|B)=P(A)
•             P(B|A)=P(B)
•             P(A AND B)=P(A)⋅P(B)
Example 1
Question
Given the following information, can we determine which pairs of A, B, and C are independent or dependent?
P(A)P(B)P(C)=0.2=0.5=0.3P(A|B)P(B|C)P(A|C)=0.5=0.5=0.2
Mutually Exclusive Events
Mutually exclusive events are even easier to identify, because mutually exclusive events cannot happen simultaneously. Therefore, if A and B are mutually exclusive,
•             P(A|B)=0
•             P(B|A)=0
•             P(A AND B)=0
Example 2
Question
Given the following information, what can we say about the relationship between events A and B?
P(A)P(B)P(B|A)=0.21=0.53=0
Example 3
Question
If P(A)=0.6 and P(B)=0.15, what can we say about the relationship between Aand B?
Key Terms
•             Independent events: events that have no influence on each other
•             Dependent events: events that influence the occurrence of the other;
•             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.
•             Conditional probability: the chance that an event will happen if another event has already happened.
QuestionGiven the following information, determine whether events B and C are independent, mutually exclusive, both, or neither.
•             P(B)=60%
•             P(B AND C)=0
•             P(C)=85%
Independent
Mutually Exclusive
Both Independent & Mutually Exclusive
Neither
QuestionThe event of using your free hour to nap is A and the event of using your free hour to study is B. If these events are mutually exclusive events, using P(A)=0.23, and P(B)=0.73, what is P(B|A)?
Question
Consider the Venn diagram below. Each of the dots outside the circle represents a university that was polled. In this experiment, 8 universities were asked about their student demographics.
•             Five of the universities are public universities: represented by red dots and labeled: 1,2,3, 4, 5 (abbreviated U);
•             Three of the universities are private universities: and are represented by blue dots: 1,2, 3 (abbreviated P).
The two events represented in the Venn diagram are:
•             Event A: The majority of the university’s students are women.
•             Event B: The majority of the university’s students are non-traditional college students.
Universities U1,U2,U3, and P1 say that the majority of their students are women. Universities P2 and P3 say that the majority of their students are non-traditional students. Place the dots in the appropriate event given the information above (you may not use all of the dots), then determine if the events are mutually exclusive.
QuestionThe event of the local baseball team winning is A and the event of data rates going up is B. If these events are independent events, using P(A)=0.17, and P(B)=0.50, what is P(A|B)?
QuestionGiven the following information about events A, B, and C.
P(A)P(B)P(C)=0.62=0.34=0.07P(B|A)P(C|B)P(A|C)=0=0.34=0.62
Are A and C mutually exclusive, independent, both, or neither?
A and C are not independent because P(A|C)≠P(A).
  A and C are independent because P(A|C)=P(A).
 A and C are mutually exclusive because P(A|C)=P(A).
A and C are both mutually exclusive and independent.
 
QuestionYou have a standard deck of 52 cards. There are four suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards: Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King. Given Events A and B, are the two events mutually exclusive? Explain your answer.
Event A: Drawing a 10.
Event B: Drawing a heart.
Yes, the events are mutually exclusive because they have no outcomes in common.
 Yes, the events are mutually exclusive because P(A) is not equal to P(B).
 No, the events are not mutually exclusive because they both require drawing a card
 No, the events are not mutually exclusive because they share the common outcome of 10 of hearts.
QuestionGiven the following information about events B and C
•             P(C|B)=38
•             P(B)=12
•             P(C)=38
Are B and C mutually exclusive, independent, both, or neither?
B and C are independent because P(C|B)=P(C).
 B and C are mutually exclusive because P(C|B)=P(C).
 B and C are neither mutually exclusive nor independent.
 B and C are both mutually exclusive and independent.