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.

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