Estadística para decisiones /Statistics for Decision Making 1
Universidad del Turabo
STAT 555 – DL WORKSHOP THREE
Topic: Probabilities
Introduction
Welcome to Workshop Three!
It’s time to learn to manage uncertainty when making decisions in a business
environment. That is why in this workshop we will begin to study the odds as a
fundamental tool to answer questions like: What are the chances that sales will
decrease if prices are increased? What is the likelihood that a new assembly
process will increase productivity? or, What is the chance that a new investment
will be profitable?. Also, probabilities can help us face situations like: your division
is just about to decide whether or not to introduce a new laptop into the consumer
market. Although your marketing studies show that typical consumers liked the
product and feel that the price is reasonable, its success is hardly assured.
Uncertainty comes from many factors: competition, suppliers, economic situation,
etc.
Specific Objectives:
At the end of the workshop, you will be able to:

Define the probability concepts and their relation to statistics.

Explain what an experiment, a sample space, and an event are.

Describe various approaches: classical, relative frequency, and subjective
probability.

Define conditional probability and joint probability.

Calculate probabilities using addition and multiplication rules.

Use a contingency table and a tree diagram to organize and calculate
probabilities.

Use Bayes’ Theorem to revise probabilities based on known information.
Language Objectives:

Analyze the situation, identify the problems, and develop solutions for the
problems in English. Answer the assigned questions in English with the
correct language structure, syntax, and the correct grammar.
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Estadística para decisiones /Statistics for Decision Making 2

Ask questions in a professional manner using strategic ideas and factors in
writing utilizing the correct terms in accordance with the course content and
using the proper rules in grammar.

Present statements and arguments showing leadership, professionalism and
considering the different opinions of the audience, in oral form with the
proper pronunciation, using the correct grammar and verbs, ensuring the
terms and statements follow the content of the course.
Course content presentation:
3.1
Basic concepts of probabilities
To understand the probabilities you need to know the following terms:
Experiment
An activity or measurement that results in an outcome.
Sample
All possible outcomes of an experiment.
space
Event
One or more of the possible outcomes of an experiment, a
subset of the sample space.
Probability
A number between 0 or 1 which express the chance that an
event will occur.
Properties:
0<P(A)<1
P(A) + P(AC) = 1
Note: Complement of an Event – given an event A, the complement of A is defined
to be the event consisting of all sample points that are not in A. The complement of
A is denoted Ac. P(A) = 1 – P(Ac)
Example:
Experiment
Next week’s trading activities on the New York Stock
Exchange
Sample
Consists of two possible outcomes: (1) the Dow Jones Industrial
space
Average goes up by at least 30 points, and (2) it does not.
Event
The Dow Jones Industrial Average goes up at least 30 points
next week.
Probability
The chance that the Dow Jones Industrial Average will increase
by at least 30 points next week.
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Estadística para decisiones /Statistics for Decision Making 3
Approaches to assign probabilities:
1) Classical approach – describes a probability in terms of the proportion
of times that an event can be theoretically expected to occur
2) Relative Frequency approach – the probability is the proportion of
times an event is observed to occur in a very large number of trials.
3) Subjective approach – the probability is a judgment, representing the
degree which one happens to believe that an event will or will not
occur.
Also, you are invited to watch the following videos about this topic
selecting the following link.
Probability Introduction
http://www.youtube.com/watch?v=Vmpqpnb1Xyg
Classical Probability
http://www.youtube.com/watch?v=RZgB5bQOfxI
Relative Frequency approach
http://www.youtube.com/watch?v=WBLcSXIboCM
These links are interesting because they explain in a simple way and with
good examples a probability, and how through the classical methods and
frequency distribution can probabilities be assigned.
I invite you to practice and observe since knowing about probabilities is useful in
the daily lives of people.
Activities
This activity does not have any points and will not be considered when assigning
the grade for the course, however, it will help you clarify any doubts and answer
any questions that you may have. Also, it will help you in getting prepared to do
the assignments and to take the short test. This will be part of the final evaluation
and you will find them at the end of the workshop.
Activity 3.1 Problems related to the basic concepts of probabilities
The following problems are related to basic concepts of probabilities. Once you
consider that you understand these concepts, solve the problems.
1. Video Game Inc., recently developed a new video game. Its playability is
to be tested by 80 veteran game players.
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Estadística para decisiones /Statistics for Decision Making 4
(a) What is the experiment?
(b) What is one outcome?
(c) Suppose 65 players tried the new game and said they liked it. Is 65
a probability?
(d) The probability that the new game will be a success is computed to
be -1.0. Comment.
(e) Specify one possible event.
2. In each of the following cases, indicate whether classical, relative
frequency, or subjective probability is used:
(a) A baseball player gets a hit in 30 out of 100 times at bat. The
probability that he gets a hit in his next at bat is 0.3.
(b) A seven-member committee of students is formed to study
environmental issues. What is the likelihood that any one of the seven
will be chosen as the spokesperson?
(c) You purchase one of 5 million tickets sold for lotto in Canada. What is
the likelihood you will win the $1 million jackpot?
(d) The probability of an earthquake in northern California in the next 10
years is .80.
3. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name:
W3.A1.name.lastname.
4. Finally, you are required to send your responses to the facilitator using
the Tareas/Tasks option on the menu, select Workshop Three/
Practice T3.A1 and Click to launch for submit.
5. After the facilitator receives your response, it will be analyzed and you
will receive recommendations and comments (if any).
STAT 555-DL
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Estadística para decisiones /Statistics for Decision Making 5
3.2
Some rules for computing probabilities
Before presenting the rules it is essential to define some terms:
Mutually
If one event occurs, the other cannot occur.
exclusive events
Exhaustive
A set of events is exhaustive if it includes all the
events
possible outcomes of an experiment.
Intersection of
Two or more events occur at the same time.
events
Representing: “A and B”
Union of events
At least one of a number of possible events
occurs.
Representing: “A or B”
Marginal
The probability that a given event will occur.
probability
Joint probability
The probability that two or more events will all
occurs.
Conditional
The probability that an event will occur, given that
probability
another event has already happened.
Representing:
P( A / B) 
P( A and B)
P( B)
Independent
Events are independent when the occurrence of
probability
one event has no effect on the probability that
another will occur.
Dependent
Events are dependent when the occurrence of one
probability
event changes the probability that another will
occur.
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Estadística para decisiones /Statistics for Decision Making 6
1. Rules of addition
Special rule of addition – to apply this rule the events must be
mutually exclusive.
P(A or B) = P(A) + P(B)
2. General rule of addition – apply when the events are not mutually
exclusive. When two events occur at the same time the probability is
called a joint probability.
P(A or B) = P(A) + P(B) – P(A and B)
3. Rules of multiplicationSpecial rule of multiplication – requires that two events A and B are
independent. Two events are independent if the occurrence of one event
does not alter the probability of the occurrence of the other event.
P(A and B) = P(A)P(B)
a. General rule of multiplication – applies when two events are
dependent. The probability is called a conditional probability.
P(A or B) = P(A)P(B/A)
Also, you are invited to watch the following video about this topic
selecting the following link.
Addition rule
http://www.youtube.com/watch?v=tdtys3kcSIk
(Part I)
http://www.youtube.com/watch?v=1dKf6pc3lgI
(Part II)
Multiplication rule
http://www.youtube.com/watch?v=YyhuySxKvu8
These links will be very helpful because they present with examples how
addition and multiplication rules apply to probabilities.
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Estadística para decisiones /Statistics for Decision Making 7
Here you will find a new activity to practice probability rules, enjoy it.
Activity 3.2 Problems related to addition and multiplication rules
The following problems are related to addition and multiplication rules. Once you
consider that you understand these concepts, solve the problems.
2.
A survey of employees at a large company found the following relative
frequencies for the one-way distances they had to travel to arrive at work:
Number of Miles (One-Way)
Relative
A
B
C
D
E
F
<5
6-10
11-15
16-20
21-30
>31
0.38
0.25
0.16
0.09
0.07
0.05
Frequency
(a) What is the probability that a randomly selected individual will have to
travel 11 or more miles to work?
(b) What is the probability that a randomly selected individual will have to
travel between 6 and 15 miles to work?
(c) Using the letter identifications provided, calculate the following
probabilities: P(A or B or E); P(A or F); P(A’ or B); P(A or B or C’).
3. A study by the U.S. Energy Information Administration found that 84.3% of
U.S. household with incomes under $10,000 did not own a dishwasher while
only 21.8% of those in an income range over $50,000 did not own a
dishwasher. If one household is randomly selected from each income group,
determine the probability that:
(a) neither household will own a dishwasher.
(b) both households will own a dishwasher.
(c)
the lower-income household will own a dishwasher, but the higherincome household will not.
(d) the higher-income household will own a dishwasher, but the lowerincome household will not.
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Estadística para decisiones /Statistics for Decision Making 8
4. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W3.A2.name.lastname.
5. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Three/Practice T3.A2
and Click to launch for submit.
6. After the facilitator receives your response, they will be analyzed and you will
receive recommendations and comments (if any).
3.3
Contingency Tables and Tree diagram
Sometimes it is important to organize the data to calculate the probabilities.
Below are two methods commonly used:

Contingency tables - a table used to classify sample observations
according to two or more identifiable characteristics. This is a crosstabulation that simultaneously summarizes two variables of interest and
their relationships.

Tree diagram – is a graph that helps organize calculations that involve
several stages. Each segment in the tree is one stage of the problem. The
branches of a tree diagram are weighted by probabilities.
Also, you are invited to watch these videos selecting the following links:
Contingency table:
http://www.youtube.com/watch?v=zEjA2n-3LFk
Tree Diagram:
http://www.youtube.com/watch?v=mkDzmI7YOx0
http://www.youtube.com/watch?v=qRgf_6CUHZI
Here is a good opportunity to appreciate an easy development of a
contingency table and tree diagram.
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Estadística para decisiones /Statistics for Decision Making 9
Here you will find other activities to practice probabilities using
contingency tables and tree diagrams.
Activity 3.3 Problems related to contingency tables and tree diagrams
The following problems are related to contingency tables and tree diagrams. Once
you consider that you understand these concepts, solve the problems.
1. Due to rising health insurance costs, 43 million people in the United States
go without health insurance. Sample data representative of the national
health insurance coverage are shown here:
Health Insurance
Age
Yes
No
18 to 34
750
170
35 and older
850
130
(a) Develop a joint probability table for these data and use the table to
answer the remaining questions.
(b) What do the marginal probabilities tell you about the age of the U.S.
population?
(c) What is the probability that a randomly selected individual does not have
health insurance coverage?
(d) If the individual is between the ages of 18 and 34, what is the
probability that the individual does not have insurance coverage?
(e) If the individual age is 35 or older, what is the probability that the
individual does not have health insurance coverage?
(f)
If the individual does not have health insurance, what is the probability
that the individual is in the 18 to 34 age group?
(g) What does the probability information tell you about health insurance
coverage in the United States?
2. A repair shop has two technicians with different levels of training. The
technician with advanced training is able to fix problems 92% of the time,
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Estadística para decisiones /Statistics for Decision Making 10
while the other has a success rate of 80%. Suppose that you have a 30%
chance of obtaining the technician with advanced training.
(a) Draw a tree diagram for this situation.
(b) Find the probability that your problem will be fixed.
(c) Given that your problem is fixed, find the probability that you did not
obtain the technician with advanced training.
3. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W3.A3.name.lastname.
4. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Three/Practice
T3.A3 and Click to launch for submit.
-----------------------------------------------3.4 Bayes’ Theorem
An important phase of probability analysis is to revise the probabilities when
new information is obtained. Often, we begin the analysis with initial or prior
probability estimates for specific events of interest. Then, from sources such
as a sample, a special report, or a product test, we obtain additional
information about the events. Given this new information, we update the
prior probability values by calculating the revised probabilities, referred to as
posterior probabilities. Bayes’ Theorem provides a means for making these
probability calculations.
Bayes' Theorem
P( A1 / B) 
P( A1 ) P( B / A1 )
P( A1 ) P( B / A1 )  P( A2 ) P( B / A2 )
Also, you are invited to watch the following video with audio about this
topic selecting the following link.
Introduction to Bayes’ theorem
http://www.youtube.com/watch?v=pPTLK5hFGnQ
This is a good example of the implementation of the Bayes’ theorem.
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Estadística para decisiones /Statistics for Decision Making 11
Concentrate and tell me, what are the results of the following problem?
Activity 3.4 Problems related to Bayes’ theorem
The following problem is related to Bayes’ theorem. Once you consider that you
understand these concepts, solve the problem.
1. For U.S. live births, P(boy) and P(girl) are approximately 0.51 and 0.49,
respectively. According to a newspaper article, a medical process could alter
the probabilities that a boy or a girl will be born.
Without medical intervention, researchers using the process claim that
couples who wanted a boy were successful 85% of the time, while couples
who wanted a girl were successful 77% of the time. Assuming that the
medical process does have an effect on the sex of the child:
(a) What is the probability of having a boy?
(b) With medical intervention, what is the conditional probability that a
couple who wants a boy will have a boy?
(c) With medical intervention, what is the conditional probability that a
couple who wants a girl will have a girl?
2. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W3.A4.name.lastname.
3. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Three/Practice
T3.A4 and Click to launch for submit.
4. After the facilitator receives your response, it will be analyzed and you will
receive recommendations and comments (if any).
STAT 555-DL
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Estadística para decisiones /Statistics for Decision Making 12
I invite you to participate in the following Forum where you can share any
thoughts or doubts you have about the topic in discussion.
To join us, access Comunicación/Communication link and press Workshop
Three: Doubts and Questions.
The assignments below will help you reflect on the discussed topic. Follow the
instructions provided for each assignment.
Tasks/Activities:
Task 3.1 Project Part A (written)
Instructions
1. Choose a specific decision problem related to your business interest that
depends on two uncertain events. For example: the introduction of a new
product will be successful or not.
2. Select reasonable initial values for three probabilities.
3. Report two relevant probabilities and two relevant conditional probabilities,
and interpret each other.
4. Write two paragraphs discussing what you have learned about this project.
5. Use MS Word or a compatible program and save it as:
T3.1.name.lastname.
6. The facilitator will use the Appendix H- Workshop 3: Project Evaluation
Rubric (15 points) for this task and you can find it in the syllabus.
7. Use MS Word or a compatible program and save as:T3.1.name.lastname.
8. Submit to the facilitator using the link Tareas/Task selecting Workshop
Three and click on T3.1: Part A Project.
Task 3.1 Part B Project Voice forum - (oral)
Instructions
In addition, participate in a voice discussion forum T3.1, Part B: Project.
1. You can review the instructions or tutorial for the voice forum accessing
e-Lab button.
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Estadística para decisiones /Statistics for Decision Making 13
2. The facilitator will use the Appendix H- Workshop 3: T3.1 Part B-Project
Discussion Forum Rubric. (5 points)
3. The duration of the recording should fluctuate between 1 and 3 minutes.
4. Please record your message through Tareas/Tasks, select Workshop
Three and press T3.1. Part B Project Voice Forum link to submit to your
colleagues the selected company, the decision data processing problem
facing, the report and the interpretation.
5. Read several of your classmate forums and react to at least two of them.
Points: The project will have a value of 20 points (T3.1.Part A -15 points;
T3.1 Part B - 5 points).
Task 3.2 Short Test 3
Instruction
1. At the end of Workshop Three, you will take a short test. You should have
studied the topics in the workshop and answered all the problems before
taking this test.
2. This test has ten (10) multiple-choice questions. To access the test, go to
the Tareas/Tasks option, select Workshop Three and click Short Test 3–
Probabilities. You will see the page with the questions and multiple
choices. Every time you answer a question, it is important to press “save”
and when you finish the test, you must press “submit”.
3. Read each question carefully; do not answer the question until you are sure
what is being asked. You will see the points received, and any incorrect
answers you had (if any) with the correct answer.
Points
The short test will have a value of 20 points; each question is worth 2 points.
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Estadística para decisiones /Statistics for Decision Making 14
Task 3.3 Voice Forum Reflective Journal (oral)
Instructions:
1. At the end of Workshop Three, you will prepare an oral work reflection
2. In this activity, you should answer the following questions:
a. Which topic did I learn?
b. How can I use it in my daily life?
c. What would you have added to the workshop?
d. Which topic impressed me the most? Why?
e. Which topic did I like the least? Why?
f. Which topic did I not understand?
3. You can review the instructions or tutorial for recording accessing e-lab.
4. The facilitator will use Appendix J- Workshop 3: Reflective Journal to
evaluate this task and you can find it in the syllabus.
5. The duration of the recording should fluctuate between 1 and 3 minutes.
6. Please record your message through Tareas/Tasks and click Workshop
Three; press T3.3 Voice Forum: Reflective Journal.
Points
The work reflection is worth 10 points.
Good Work, you have finished Workshop Three!
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