DEVRY MATH399 Week 3 Assignment Independent and Mutually Exclusive Events Latest 2019 JULY Question # 00603572 Subject: Mathematics Due on: 08/10/2019 Posted On: 08/10/2019 05:37 AM Tutorials: 1 Rating: 4.7/5

MATH399 Applied Managerial Statistics

Week 3 Assignment Independent and Mutually
Exclusive Events

Question Given
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.