Lesson 1: (On Campus) Course Intro & Probability

Optional Video for this Lesson

Lesson Outcomes

By the end of this lesson, you should be able to:

1. Explain the course policies
2. Access course resources (course outline, lesson schedule, preparation activities, reading quizzes, homework assignments, assessments, etc.)
3. Communicate with the instructor and group members
4. Access statistical analysis software tools for class quizzes, assignments, and exams
5. Apply principles of the gospel of Jesus Christ in this class
6. Apply the three rules of probability for different probability scenarios

Welcome to the Course!

In this course, you will explore important connections between the academic discipline of Statistics and the world around us. By pondering these ideas, your understanding of statistics will increase, as will your knowledge and testimony of the restored Gospel of Jesus Christ. In addition, that which you learn in this course will increase your ability to serve others as a disciple of Jesus Christ and help build Zion.

This course has been designed to help you slowly build up a knowledge base of ideas and skills. Not all of these ideas and skills will come easily. It takes a lot of work and practice before some things will even start to make sense, so you should not be surprised to find that it may take you a little time to comprehend these ideas. Just be patient. Once you’re far enough into the course, the ideas will start to come together, and you will see how much progress you have really made. You will understand what this course is all about, and you will be glad you persisted in your efforts to learn.

Course Description

This course covers the following topics as they are applied to Statistics: graphical representations of data, measures of center and spread; elementary probability; sampling distributions; correlation and regression; statistical inference involving means, proportions, and contingency tables.

Course Learning Outcomes

In this course, we will:

1. Summarize data numerically and graphically using spreadsheets
2. Make decisions regarding situations with inherent randomness
3. Apply probability distributions to investigate questions
4. Employ confidence intervals in various situations
5. Implement tests of diverse hypotheses
6. Communicate the results of statistical analyses to relevant audiences

How the Outcomes will Be Assessed

While you may not be tested on everything you learn in this course, the instructor will be assessing your mastery of the Course Learning Outcomes. The general types of assessments used to measure these outcomes may include selected response tests such as multiple-choice, true-false, matching, and fill-in-the-blank questions. You may also be asked to complete essays or other writing assignments. At times, the instructor may assess your performance of a skill, or the instructor may assess products you create using particular skills. In addition, the instructor may engage in personal communication with you to determine how well you understand the course content.

Keys to Success

Five Principles of the Learning Model

You will experience much deeper learning if you follow the Five Principles of the BYU-Idaho Learning Model

• Exercise Faith: Exercise faith in the Lord Jesus Christ as a principle of action and power.
• Learn by the Holy Ghost: Understand that true teaching is done by and with the Holy Ghost.
• Lay Hold on the Word of God: Lay hold of the word of God.
• Act for Themselves: Act for yourself and accept responsibility for learning and teaching.
• Love, Serve, and Teach One Another: Love, serve, and teach other students in your classes.

Three Process Steps of the Learning Model

You will learn more in less time if you follow the Three Process Steps of the BYU-Idaho Learning Model

• Prepare: This involves (a) spiritual preparation, (b) individual preparation, and (c) group preparation.
• Teach One Another: You should (a) be on time, (b) pray together, and (c) actively engage with other students.
• Ponder/Prove: You should (a) ponder what you have learned, (b) record your learning, and (c) pursue unanswered questions and discuss what you learn with others.

If you feel confused or have questions about anything in the lesson, take immediate action (Exercise Faith; Act for Themselves) and talk with your classmates, the teaching assistant, or the instructor (Love, Serve, and Teach One Another).

Teach One Another

At BYU-Idaho, an “A” student will demonstrate “diligent application of Learning Model principles, including initiative in serving other students” (BYU-Idaho Catalog). In this class, you will have the opportunity to work with other students.

Doctrine and Covenants 84:106 states, “And if any man among you be strong in the Spirit, let him take with him him that is weak, that he may be edified in all meekness, that he may become strong also.” In the spirit of this revelation, you will have the opportunity to help others in the class when you have developed an understanding of a principle. Likewise, you will be able to receive help from others (peers, tutors, TA, and your instructor) when you are still working to understand concepts.

In a spirit of love and service, please reach out to others. You are not graded on a curve. If someone else does well, it does not affect you adversely. Research has shown that students who help other students to understand the material gain a much deeper grasp on the concepts of the course. Please take opportunities to help your peers succeed.

Course Structure

This course consists of 24 lessons. They are presented in a topical order in which concepts and skills learned in the earlier lessons provide the requisite knowledge to succeed in later lessons. If the general order of the lessons doesn’t make sense at first, don’t worry. It will all come together in the end, and you’ll see the reasoning behind why the lessons have been presented in this particular order.

Your main goal as a student will be to complete all of the learning activities within each lesson by their due dates every week. These activities follow a consistent weekly schedule, and it will be up to you to make sure that you keep on pace with all your assignments. These weekly activities may include the following:

• Reading assigned texts or viewing presentations.
• Taking quizzes.
• Participating in group discussions and assignments with other class members.
• Writing papers and/or developing presentations.
• Participating in meetings with the instructor, teaching assistants, and other students.

For many of these activities, the due dates will fall on the same time each week. This will make it easier for you to plan out your weekly study schedule. However, there may be a need to make adjustments to the schedule from time to time. If in doubt, refer to the due dates your instructor has posted in I-learn.

You should create a study schedule that will keep you on pace throughout the semester. This is a rigorous course with a lot of subject matter to cover, and it can be extremely difficult to recover if you fall too far behind in your work. So, please make every effort to study on a regular basis and get your work turned in on time.

The lessons in this course have a similar structure and contain similar basic elements. A typical week consists of two lessons. Each lesson will consist of a reading assignment, a reading/preparation quiz, and a homework quiz.

The structure of this course fully integrates the BYU-Idaho Learning model with a mixture of preparation activities, teach-one-another activities, and ponder-and-prove activities.

Course Materials

This course has been designed with the student in mind. Every effort has been made to provide a high quality experience at the lowest possible cost.

Textbook

To keep costs as low as possible for students and their families, no physical textbook is required for this class. The readings for this course are provided on this website and will continue to be available to you after the course is completed. Please report any problems with the textbook (links not working, loading slowly, inability to view images, etc.) to your instructor. A link to the textbook is found in the Quick Links module. It is highly recommended you bookmark the textbook so that you can easily reference each lesson’s reading.

Computer Equipment

You will need: - A laptop - Access to Microsoft Excel 2016 or later

Course Resources

Peer Support

Your experience in this course will be enhanced as you work with other students to learn and grow together.

Help Desk

The BYU-Idaho Help Desk has been established to help students with technological problems related to approved course software. You can access the Help Desk at any time in three ways: - Walk-in: The Help Desk is located in room 322 of the McKay Library - Call in: 208-496-1411 (toll free) - Email: helpdesk@byui.edu Additional information is available at the Help Desk web page: http://www.byui.edu/helpdesk/

When you have technical problems with I-Learn, you should first try contacting the Help Desk before you contact your instructor. They are connected with the IT support staff who can resolve problems with I-Learn. Please take a moment now to look at the Help Desk web page. That way, if a problem does arise later on in the course, you will know where to go for help.

Tutoring Center

The BYU-Idaho Study Skills/Tutoring Center is a powerful resource for students who would like a little extra help with a course. The Tutoring Center is located in the McKay Library in room 272. This is in the east wing of the second floor.

The Tutoring Center provides many services to help students succeed: - Individual tutors - Walk-in tutoring in the Math Study Center (McKay 266 & 270) - Virtual tutoring

Please take 5 minutes to explore the Study Skills/Tutoring Center web site.

Faculty Support

Your instructor is committed to your success. If you have any needs or concerns, please contact your instructor for help. If you feel yourself getting behind or struggling, talk to your teacher right away. If caught in time, a small problem can be addressed quickly before it grows.

With all of that said, let’s begin looking at a foundational idea of statistics: probability.

Probability

Probability is a way of numerically quantifying how likely an event is to happen or not happen. The following historical account demonstrates this idea and shows how fractions (like 1/2 or 3/4) or percentages (like 50% or 75%) can be used to represent probabilities.

Christopher Columbus’ First Voyage

On August 3, 1492, Columbus set sail from Spain for his intended destination: the Indies (Caso, Adolph 1990). He was on the Santa Maria, which had a crew of approximately 41 men (“Cristobal colon” 1991; “Christopher Columbus”). Several other men were aboard the Nina and the Pinta (“Cristobal colon” 1991). On October 12, he landed on an island in the Bahamas he called San Salvador.

The return trip was not without challenges. The Santa Maria ran aground on Christmas Day, 1492, and was abandoned on the island we now call Hispaniola (home to Haiti and the Dominican Republic). Following this incident, Columbus sailed for Spain. Severe storms made the journey difficult. A particularly bad storm on February 14, 1493 made the crew fear for their lives. By morning, the storm was even worse!

Recognizing his dependence upon God, Columbus ordered that a pilgrimage should be made to a particular shrine upon their safe arrival in Spain. He decided that they would use random chance to determine who would make the pilgrimage. They took one chick pea for each man on board. A knife was used to mark one of the chick peas with a cross. The chick peas were placed in a hat and shaken up. Each man was to draw a chick pea, and the one who had the cross would make the pilgrimage.

“The first who put in his hand was [Columbus,] and he drew out the bean with a cross, so the lot fell on him; and he was bound to go on the pilgrimage and fulfil the vow” (Caso, Adolph 1990).

Answer the following questions:

1. Remember, there were 41 men aboard his ship. What is the probability that Columbus would draw out the marked chick pea? Express your answer as a fraction, and then convert it to a decimal.

Show/Hide Solution

• There is only one marked chick pea in the hat, out of 41 chick peas total. Out of is expressed arithmetically by division. The probability is \(\frac{1}{41} = 0.0244\). (Note: this is about 2%.)

2. Based on your answer to the previous question, how likely is it that Columbus would draw out the marked chick pea?

Show/Hide Solution

• There is only about a 2% chance that Columbus will draw out the marked Chick Pea. This is not very likely.
  

 

A Second Drawing
Columbus’ promise to make the pilgrimage did not stop the storm. It was determined that there should be a pilgrimage to another site they held sacred. Again, chick peas representing each member of the crew were placed in a hat and shaken up. The lot fell on a sailor…named Pedro de Villa (Caso, Adolph 1990).

Answer the following questions:

3. What is the probability that Columbus would not draw out the marked chick pea? Express your answer as a fraction, and then covert it to a decimal?

Show/Hide Solution

• There are 40 other men on board plus Columbus. So, the probability that Columbus would not draw out the marked chick pea is: \(\frac{40}{41} = 0.9756\). (Note: this is almost 98%.)

4. Based on your answer to the previous question, how likely is it that Columbus would not draw out the marked chick pea?

Show/Hide Solution

• It is very likely that Columbus would not draw out the marked chick pea. This result is not surprising.

5. In this second drawing, either Columbus would draw out the marked chick pea, or he would not. Add the probability that Columbus would draw out the marked chick pea and the probability that he would not draw out the marked chick pea. What is the value of this sum?

Show/Hide Solution

• The sum of the probabilities is 1:

\[ \frac{1}{41} + \frac{40}{41} = \frac{41}{41} = 1 \]

 

Additional Drawings

After the drawing in which Pedro de Villa was chosen to make a pilgrimage, two additional drawings were held. In both cases, Columbus drew out the marked chick pea (Caso, Adolph 1990). In all, Christopher Columbus drew the marked chick pea in three of the four drawings. It can be shown that the probability that this would occur due to chance is very small: 0.0000566. (Show/Hide Solution)

Bonus material. Read only if you are interested.

This calculation is more involved than the calculations you will be required to make in this course this semester. But if you are still interested, read on.

In each individual drawing, there was a 1/41 chance of Columbus getting the marked chick pea. Similarly, there was a 40/41 chance of not getting it. Since there were four drawings total, and the goal is to measure the probability of “three of those drawings” resulting in Columbus getting the marked chick pea, it becomes important to think about all of the orders in which Columbus could have gotten 3 out of 4.

Possible Outcome	First Drawing	Second Drawing	Third Drawing	Fourth Drawing
What actually happened…	Got it.	Didn’t get it.	Got it.	Got it.
But he could have…	Got it.	Got it.	Didn’t get it.	Got it.
Or he could have…	Got it.	Got it.	Got it.	Didn’t get it.
Or he could have…	Didn’t get it.	Got it.	Got it.	Got it.

In each of the above cases, notice that Columbus would have gotten the marked chick pea a total of 3 out of 4 times. So, this tells us there are four diffent ways to get the chick pea 3 out of 4 times.

The probability of what actually happened to Columbus in the order in which it happened would be computed by multiplying the individual probabilities of each drawing together.

\[ \frac{1}{41} \cdot \frac{40}{41} \cdot \frac{1}{41} \cdot \frac{1}{41} \approx 0.00001415548 \]

But then, we must also add to this the other “possible” scenarios that would also lead to getting the chick pea 3 out of 4 times, but as shown below, because multiplication is commutative (the order doesn’t matter) these “different” situations result in the same probability as the first.

\[ \frac{1}{41} \cdot \frac{1}{41} \cdot \frac{40}{41} \cdot \frac{1}{41} \approx 0.00001415548 \]

\[ \frac{1}{41} \cdot \frac{1}{41} \cdot \frac{1}{41} \cdot \frac{40}{41} \approx 0.00001415548 \]

\[ \frac{40}{41} \cdot \frac{1}{41} \cdot \frac{1}{41} \cdot \frac{1}{41} \approx 0.00001415548 \]

Thus, all that is needed is to multiply the first probability of roughly 0.00001415548 by 4 to get \(0.00001415548 \cdot 4 = 0.00005662192\).

End of Bonus Material.

To put some perspective on this, a high school athlete in the United States is over 26 times more likely to play professional sports than Columbus was to draw the three marked peas! (Fields, Mike 2008) Consider how you might explain the occurrence of this extremely unlikely event. (While no response is required of you right now, this kind of question will be very important later, so take a little time to ponder it.) In fact, it is worth restating the question, “How might you explain the occurrence of this extremely unlikely event?”

Now, take a moment to practice what you have read by answering the following questions.

Answer the following questions:

6. If a fair, six-sided die* is rolled, what is the probability of rolling a 6?

Show/Hide Solution

• The probability of rolling a 6 on a die is \(\displaystyle{ \frac{1}{6}} = 0.1667\). This is because a six-sided die has 6 sides total and only 1 side that has a “6” on it. Thus, 1/6 represents the chance of getting a “six” divided by the “total number of possibilities on the die” in the denominator (6).

7. If a fair, six-sided die is rolled, what is the probability of not rolling a 6?

Show/Hide Solution

• The probability of not rolling a 6 on a die is \(\displaystyle{\frac{5}{6}} = 0.8333.\) This is because there are five sides on the die that are “not a 6” (the sides representing 1, 2, 3, 4, and 5) and six total sides, so 5 out of 6 sides will yield something other than a “six.”

8. When a die is rolled, what is the sum of the probability of rolling a 6 and the probability of not rolling a six?

Show/Hide Solution

• The probability of rolling a six is \(\displaystyle{\frac{1}{6}}\) and the probability of not rolling a six is \(\displaystyle{\frac{5}{6}}\). These things cannot happen at the same time, so the probability of either rolling a six or not rolling a six is \[ \frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1 \]

*The only possible outcomes in this case are that you either roll a six or that you do not roll a six. The probability that one of these will happen is 1. If we list all the possible outcomes, the probability that at least one of them will occur is 1.

9. In general, if we know the probability that a particular outcome will occur, how could we calculate the probability that it will not occur?

Show/Hide Solution

• If we know the probability that an event will occur, we can subtract this probability from 1 to find the probability that the event will not occur. This is because the sum of the probabilities that the event will occur and that the event will not occur must be 1.

\(*\)Note: The word “die” is the singular form of the word “dice.” When we refer to a die, we are talking about a fair, six-sided die.

Probability Notation

You may already have a good understanding of the basics of probability. It is worth noting that there is a special notation used to denote probabilities. The probability that an event, \(x\), will occur is written \(P(x)\) and pronounced as “probability-of-event-x.” As an example, the probability that you will roll a 6 on a fair six-sided die can be written as

\[ P\text{(Roll a "six" on a fair six-sided die)}= \frac{1}{6} = \frac{\text{number of sides that show a "six"}}{\text{total number of sides on the die}} \]

Rules of Probability

Probabilities follow patterns, called probability distributions, or just distributions, for short. There are three rules that a probability distribution must follow. Answer the following questions to explore what these three rules might be.

Answer the following questions:

10. As an answer to a homework problem, a student reported, The probability of finding life on Mars is -0.35. What is wrong with this statement?

Show/Hide Solution

• Probabilities cannot be negative because they represent the “number of ways something specific can happen” (like how many planets that are “just like” Mars and do have life on them) divided by “the total number of possibilities” (in this case, the total number of planets that exist that are “just like” Mars). The fewest planets there could be that are “just like” Mars and have life on them is zero. So the lowest a probability can go is zero.

11. A student in an introductory statistics class wrote the following statement on an exam: The probability that the event will occur is 1.25 (i.e. 125%). What is wrong with this statement?

Show/Hide Solution

• Probabilities cannot be larger than 1 (or 100%). This is because probabilities represent frequencies of occurrence, and the most something can happen is “all the time” or 100% of the time.

12. A student estimated that the probability that he would finish his homework is 0.45 (i.e., 45%), and the probability that he would not finish his homework is 0.35 (i.e., 35%). What is wrong with this student’s statement?

Show/Hide Solution

This can be viewed as one of two problems:

1. The probabilities for all the events do not add up to 1 (or 100%.)

2. The probability that he does not finish his homework is actually 1 minus the probability that he will finish his homework or 0.55 (i.e., 55%).

 

In this course we are interested in experiments where the outcomes of the experiment are uncertain, yet they follow a pattern or probabilitiy distribution. As you read in the above questions and answers, these probability distributions follow three rules.

  The three rules of probability are:

• Rule 1: The probability of an event \(X\) is a number between 0 and 1.

\[0 \leq P(X) \leq 1\]

• Rule 2: If you list all the outcomes of an experiment (such as rolling a die) the probability that one of these outcomes will occur is 1. In other words, the sum of the probabilities of all the possible outcomes of any experiment is 1.

\[\sum P(X) = 1\]

• Rule 3: (Complement Rule) The probability that an event \(X\) will not occur is 1 minus the probability that it will occur.

\[P(\text{not}~X) = 1 - P(X)\]

  You may have noticed that the Complement Rule is just a combination of the first two rules.

Answer the following questions:

13. Which of the probability rules was violated by the statement in Question 10?

Show/Hide Solution

• Rule 1

14. Which of the probability rules was violated by the statement in Question 11?

Show/Hide Solution

• Rule 1

15. Which of the probability rules was violated by the statement in Question 12?

Show/Hide Solution

• Rule 2 or Rule 3

 

Informally, a distribution can be thought of as being “all the possible outcomes of an experiment and how often they occur.”

Randomness

A BYU-Idaho student was overhead saying, “I went shopping and bought some random items.” Did the person actually take a random sample of the items at the store? Did they write all the items down and randomly select the items for purchase? Of course not!

What did the student mean? That the items they bought seemed unrelated. When we consciously or subconsciously choose a sample, it is not random.

So, what does it mean to be random? When something is random, it is not just haphazard, with no pattern. A random process follows a very distinct pattern over time—the distribution of its outcomes. For example, if you roll a die thousands of times, about one-sixth of the time you will roll a four. This is a very clear pattern, or part of a pattern. The entire pattern (or, the entire distribution) is that each number on the die is rolled about one-sixth of the time.

But there’s something different about the patterns followed by random processes than other kinds of patterns. Other kinds of patterns can be very predictable, such as a color pattern of the red, yellow, blue, red, yellow, blue, and so on. If you’re following this pattern and happen to see yellow, you know the next color will be blue. By contrast, you never know what you will get on the next roll of a six-sided die. You do know that in the long run you will roll fours about one-sixth of the time.

When something is random, we can be sure that it follows a long-term pattern. This long-term pattern is called its probability distribution. However, what makes “randomness” interesting is that despite knowing the long-term pattern, or how often something will occur over time, we still never know what the outcome of the next experiment will be.

Conclusion

As with all the classes you take at BYU-Idaho, it is up to you to decide what you want to get out of this class. If you choose to approach the things you study in class with an open mind, if you prepare diligently and work hard to complete all the learning activities, and if you humbly seek the Lord’s help to understand the intellectual and spiritual truths discussed in this course and in other courses, you will have an outstanding educational experience that will be a blessing to you throughout your life. May you enjoy the journey this semester into statistics!

Summary

Remember…

1. In this class you will use the online textbook that has been written for you by your statistics teachers. All of the assignments and quizzes, available in I-Learn, will be based on the readings, so study it well. Most weeks will cover two lessons.
2. You have successfully located the online textbook. Ensure you have also located the course in I-Learn and can access the quizzes and assignments that are there.
3. Ensure you have located the contact information for your instructor in the I-Learn course. Recording the contact information of peers from class would also be a wise idea.
4. This course uses MS Excel for all statistical analysis. Check that you have access to the software on your computer. If not, see I-Learn for details on how to obtain it through the University for free.
5. By doing the work, staying on schedule, and living the Honor Code you will succeed in this class.
6. The three rules of probability are:
   1. A probability is a number between 0 and 1. \[0 \leq P(X) \leq 1\]
   2. If you list all the outcomes of a probability experiment (such as rolling a die) the probability that one of these outcomes will occur is 1. In other words, the sum of the probabilities in any probability is 1. \[\sum P(X) = 1\]
   3. The probability that an outcome will not occur is 1 minus the probability that it will occur. \[P(\text{not}~X) = 1 - P(X)\]

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