Estadística Para Decisiones /Statisticsfor Decision Making 1

Estadística para decisiones /Statisticsfor 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.
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:

3.1Basic 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 space / All possible outcomes of an experiment.
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:
0P(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 space / Consists of two possible outcomes: (1) the Dow Jones Industrial 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.

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.

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.

Video Game Inc., recently developed a new video game. Its playability is to be tested by 80 veteran game players.

(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.

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.

Remember, you must submit your answers using Excel, Word or another compatible program and save it with this name: W3.A1.name.lastname.
Finally, you are required to send your responses to the facilitator using the Tareas/Tasksoption on the menu, select Workshop Three/Practice T3.A1 and Click to launchfor submit.
After the facilitator receives your response, it will be analyzed and you will receive recommendations and comments (if any).

3.2Some rules for computing probabilities

Before presenting the rules it is essential to define some terms:

Mutually exclusive events / If one event occurs, the other cannot occur.
Exhaustive events / A set of events is exhaustive if it includes all the possible outcomes of an experiment.
Intersection of events / Two or more events occur at the same time.
Representing: “A and B”
Union of events / At least one of a number of possible events occurs.
Representing: “A or B”
Marginal probability / The probability that a given event will occur.
Joint probability / The probability that two or more events will all occurs.
Conditional probability / The probability that an event will occur, given that another event has already happened.
Representing:
Independent probability / Events are independent when the occurrence of one event has no effect on the probability that another will occur.
Dependent probability / Events are dependent when the occurrence of one event changes the probability that another will occur.

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)

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)

Rules of multiplication-

Special 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)

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)

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.

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)
A
5 / B
6-10 / C
11-15 / D
16-20 / E
21-30 / F
31
Relative
Frequency / 0.38 / 0.25 / 0.16 / 0.09 / 0.07 / 0.05

(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’).

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 higher-income household will not.

(d)the higher-income household will own a dishwasher, but the lower-income household will not.

Remember, you must submit your answers using Excel, Word or another compatible program and save it with this name: W3.A2.name.lastname.
Finally, you are required to send your responses to the facilitator using the Tareas/Tasksoption on the menu, select Workshop Three/Practice T3.A2and Click to launch for submit.
After the facilitator receives your response, they will be analyzed and you will receive recommendations and comments (if any).

3.3Contingency 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 cross-tabulation 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.

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.

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?

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, 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.

Remember, you must submit your answers using Excel, Word or another compatible program and save it with this name: W3.A3.name.lastname.
Finally, you are required to send your responses to the facilitator using the Tareas/Tasksoption on the menu, select Workshop Three/Practice
T3.A3 and Click to launch for submit.

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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.

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.

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?

Remember, you must submit your answers using Excel, Word or another compatible program and save it with this name: W3.A4.name.lastname.
Finally, you are required to send your responses to the facilitator using the Tareas/Tasksoption on the menu, select Workshop Three/Practice
T3.A4and Click to launchfor submit.
After the facilitator receives your response, it will be analyzed and you will receive recommendations and comments (if any).

Tasks/Activities:

Task 3.1 Project Part A(written)

Instructions

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.
Select reasonable initial values for three probabilities.
Report two relevant probabilities and two relevant conditional probabilities, and interpret each other.
Write two paragraphs discussing what you have learned about this project.
Use MS Word or a compatible program and save it as: T3.1.name.lastname.
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.
Use MS Word or a compatible program and save as:T3.1.name.lastname.
Submit to the facilitator using the link Tareas/Task selecting Workshop Three and click on T3.1: Part A Project.

Task 3.1 Part BProject Voiceforum- (oral)

Instructions

In addition, participate in a voice discussion forum T3.1, Part B: Project.

You can review the instructions or tutorial for the voice forum accessing
e-Labbutton.
The facilitator will use the Appendix H- Workshop 3: T3.1 Part B-Project Discussion Forum Rubric. (5 points)
The duration of the recording should fluctuate between 1 and 3 minutes.
Please record your message through Tareas/Tasks,select Workshop Three and pressT3.1. Part B Project Voice Forum linkto submitto your colleagues the selected company, the decision data processing problem facing, the report and the interpretation.
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

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.
This test has ten (10) multiple-choice questions. To access the test, go to the Tareas/Tasks option, select Workshop Three and clickShort 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”.
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.

Task 3.3Voice ForumReflectiveJournal (oral)

Instructions:

At the end of Workshop Three, you will prepare anoral work reflection
In this activity, you should answer the following questions:
Which topic did I learn?
How can I use it in my daily life?
What would you have added to the workshop?
Which topic impressed me the most?Why?
Which topic did I like the least? Why?
Which topic did I not understand?
You can review the instructions or tutorial for recording accessing e-lab.
The facilitator will use Appendix J- Workshop 3: Reflective Journaltoevaluate this task and you can find itin the syllabus.
The duration of the recording should fluctuate between 1 and 3 minutes.
Please record your message through Tareas/Tasks and clickWorkshop Three; press T3.3 Voice Forum: Reflective Journal.

Points

The work reflection is worth 10 points.

Good Work, you have finished Workshop Three!