Lesson01.Rmd

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#### Optional Video for this Lesson

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## 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

<br>

## 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.

If you have never taken an online class before, you are in for a new and exciting learning experience. This course uses some of the most advanced educational tools. These technologies will help you learn in an effective and efficient way, often allowing you to learn more than you would in a traditional Statistics class---and in less time. Although some of the learning activities will require you to work on your own, you will also be engaged in activities with other students and your instructor. Through your group interactions, you will have many opportunities to teach one another by sharing ideas and solving problems together.

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.

As you thoughtfully prepare, teach one another, ponder/prove what you have learned, and as you humbly seek the guidance of the Holy Spirit, the Lord will bless you with a greater knowledge of His mercy and love. You will have the opportunity to increase your testimony of the Gospel of Jesus Christ. You will also learn about Statistics, more deeply and permanently than you would have otherwise.

<br>

## 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.

<br>

### 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


<br>

### 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.

<br>

### 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](http://www.byui.edu/LearningModel){target="_blank"}

-  **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](http://www.byui.edu/LearningModel){target="_blank"}

-  **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.

This course represents a collaborative effort between faculty, TA's, tutors, students and the Lord.  You are encouraged to seek the help from seen and unseen resources as you grapple with the principles of Statistics.

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.

<br>

## Course Structure

The structure of this course is based on scientific research on student learning and retention.  The activities, structure, pace, and repetition are all grounded in modern educational theories and practices.  This course has been designed with you, the student, in mind.

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 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. 

And now for a word of warning. If you start to fall behind, this course has built-in automatic “early warning signals” that will alert your instructor to possible problems. If your instructor contacts you to check on your progress, it is most likely because you've fallen behind in your assignments and triggered an “early warning signal.” If this happens, please work with your instructor. You’ll need to create a study schedule that will catch you up and 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.
<br>

### Lesson Pattern

The lessons in this course have a similar structure and contain similar basic elements.  A typical week consists of two lessons.

The learning activities fully integrate the BYU-Idaho Learning model.  They contain a mixture of preparation activities, teach-one-another activities, and ponder-and-prove activities. You will need to complete these activities in the order in which they are listed.

#### Individual Preparation

Each lesson begins with a reading assignment.  These readings are provided in this e-textbook.  Be sure to answer the questions in the textbook, since these are an important part of the learning process.  A note page is provided for each reading which you should use to write down the most important concepts from the reading.  Taking good notes is an important part of being prepared for future quizzes and exams.  The reading assignment must be completed before attending Group Preparation.

#### Group Preparation

Group Preparation is an opportunity to share what you have learned from Individual Preparation with your group and to learn from one another. This will help to solidify the concepts presented in the reading. Group Preparation is held twice a week at the same time of day as your class meeting. The Group Preparation session provides opportunities to work in small groups with specially trained TAs. You may choose to meet with your group via [[#Zoom Video Conferencing | Zoom]] or in person.  If you meet in person, one group member will need to be on [[#Zoom Video Conferencing | Zoom]] in order to communicate with the TAs. One member of your group will be chosen to be the Group Leader.  He or she will be responsible for reporting group attendance as well as submitting questions from the group to the TAs which will be communicated to the instructor.  

Please purchase a headset with a microphone and earphones.  You will use these to participate via [[#Zoom Video Conferencing | Zoom]]. Make sure you arrive a little early to each session in order to have time to join the meeting and set up your microphone and headphones.

As a group you will complete the practice problems provided in Part 2 of your Reading Notes and will submit your answers to the TAs for feedback. When you have successfully completed the practice problems and received feedback you may begin the Group Quiz.  While you are encouraged to work together on this quiz, each member of the group must submit his or her own answers in order to receive credit.  Attendance to Group Preparation is a requirement and will be reported to the TAs via the Group Leader. No credit will be given for the group assignment or quiz if you do not attend.

#### Homework

The homework is divided into two parts: the homework assignment and the homework quiz. Complete the homework assignment before beginning the quiz. This assignment comes with an answer key so that you may check your work. You are encouraged to collaborate with other students on the homework assignment, but it is important to the learning process that you do your own work. Before working with someone else on the assignment, you should attempt each problem on your own. After you have completed the homework assignment you may begin the homework quiz. You may use your notes, but you should complete the homework quiz without help from others. The quiz will have questions similar to the ones found in the homework assignment, so it is important that you understand well how to work through the problems in the homework assignment before beginning the quiz.

### Benefits of Lesson Schedule

You can use the calendar feature in I-learn to see the schedule of lessons. A typical week consists of two lessons.

There are several benefits of this consistent lesson structure. First, it helps you establish a regular schedule for studying and participating in class activities. Second, it helps you learn more in less time. By the third or fourth lesson, you will find that it takes you less time to navigate through the course and complete certain activities and you will know what you need to do to succeed. Third, it gives you an opportunity to develop discipleship and leadership by teaching other students on a regular basis through class meetings and small group activities. Finally, the lesson structure makes it easier for you to give feedback to the instructor so that future lessons may be improved. This is an important part of the "teach one another" concept in the BYU-Idaho learning model. By giving feedback on what does and does not work for you in each lesson, you help other BYU-Idaho students get more out of the class. So, as you go through each lesson, feel free to make notes about the things that worked well for you, as well as the things that could be improved. Include these ideas in the Lesson Evaluation Survey. Your efforts to provide feedback to the instructor will not only help improve the course, but by taking the time to think about these things, you will become a better teacher yourself.

<br>

## 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.

To keep costs as low as possible for students and their families, no textbook is required for this class.  The readings for this course are provided in this online textbook.

**Computer Equipment**

You will need:
-  A laptop
-  Access to Microsoft Excel 2016 or later
-  A headset with a microphone and earphones is also recommended

**Wiki**

The readings for this course will all be presented in this wiki.  Take some time to explore it.  You can click on [[Main Page]] in the navigation menu (upper left part of the page) to get to the main directory.
<!--
Please make an account, so you can get credit for your submissions to the wiki. To set up an account, do the following:

**Step 01**: Click on [http://byui-fdmat-wiki.pagodabox.com/index.php?title=Special:UserLogin&returnto=Main+Page Log in / create account] in the upper right-hand corner of this page.

**Step 02**: Click on "Create an account".

**Step 03**: Choose a user name and password.
*  <span style="color:red">CAUTION:</span> Do not use your BYU-Idaho (or any other important) password!  This page is not secure.  The information you submit can be viewed by others as it is transmitted.  The company hosting this wiki is working on this issue, but it has not yet been resolved.

**Step 04**: Be sure to give your real name, so you can receive the appropriate points for work you do in the wiki.

**Step 05**: Click "Create account"
-->

##### Zoom Video Conferencing

We will use the software package ''Zoom'' to hold the online course meetings.  Zoom is a software package that allows for virtual meetings, chat, and white board discussions.  Once you get used to its functionality, it will be a great tool for learning.

<br>
<br>
<div class="message Note">Take a few minutes now to familiarize yourself with the [Zoom tool](http://byu-idaho.screenstepslive.com/s/17057/m/72167/l/787891-how-do-i-get-started-with-zoom){target="_blank"}</div>
<br>
<br>


## 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 5 minutes 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)

Please take 5 minutes to explore the [http://www.byui.edu/academic-support-centers/tutoring-center Study Skills/Tutoring Center web site].

**TA Support**

In addition to the faculty who teach this course, specially trained teaching assistants (TAs) will help you through the course.  The Guided Practice sessions will be led by TAs, under the direction of the faculty. The TAs will work with you on specific discovery activities in the Guided Practice sessions.

**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.

<br>

## Plan to Succeed

<!--**How to Succeed in This Course**-->
There are three basic things you must do to succeed in this course.

1. Do the work.
2. Stay on schedule.
3. Follow the Honor Code.

If you do these three things, you will not only get a passing grade for the course, but you will have an excellent educational experience. Here's why.

#### Do the Work

This is a rigorous course. To get a passing grade, you will have to put forth a good deal of effort to complete all of the assignments. When you see a time estimate next to a learning activity, that is the approximate amount of time it will take IF you are focused on the assignment and IF you are not allowing yourself to be distracted by other things. You will have a very difficult time completing these assignments if you do them at the same time you watch TV, cook food, talk with your roommates or family members, and send text messages to your friends every three minutes. You must set aside sufficient time to do the work throughout the week (you can't do it all in one night!).  And you must find a place to do the work where you will not be distracted by anything (you can't get it done with a cell phone call interrupting you every few minutes!). Then you must commit to focus your attention on the work and do your very best to complete it in the amount of time you have scheduled. If you do this, you will learn more in less time, and you will find the work much more pleasurable.

#### Stay on Schedule

Once you have set up a regular study schedule, keep it. One of the biggest problems that students have with their courses is that they procrastinate doing the work. They put it off until they have so much work to do that they cannot figure out a way to get it all done before the due date arrives. In this course, you cannot do this and still expect to receive a passing grade. There are assignments that must be completed by specific days and times every week. If you put off doing the work until just before an assignment is due, you will find it much more difficult to concentrate on getting the work done. To succeed in this class, you will have to follow your regular study schedule every week to ensure that you do your best work and that it gets done on time.

#### Follow the Honor Code

Although there are several types of group learning activities in every lesson, this course requires a lot of individual study and work. In particular, most assignments involving a study guide and quiz are meant to be learning activities you complete on your own. If you decide to work with others to try find all the quiz answers without having to do all the assigned work, you are not only cheating yourself out of an education, you are cheating your way to a failing grade. There are other assignments throughout the course that will require you to recall information or use skills that you should have developed as you completed the learning activities. If you don't do the work, and you try to cheat your way to higher quiz grades, you will hinder your ability to complete these other assignments. Most likely, you will fail those assignments. So, do yourself and others a favor--do not cheat on any assignments, no matter how tempting it might be. Instead, renew your commitment to follow the BYU-Idaho Honor Code, double your efforts to do the work, and follow your study schedule.

#### Plan for Success

Clearly, if you want to succeed in this course, you need to figure out how you will do the work and how you will stay on schedule. Doing this will help you avoid situations in which you might be tempted to violate the Honor Code and cheat. Such situations can arise quicker in the semester than you might think. So, it is important for you to start making a plan for success right away.  Take time now to write your own plan for success. This plan should include a weekly study schedule and a general list of things you think you will need to do each week to get the work done. Be prepared to discuss this plan with your study group and with the instructor.

-  Write your own Plan for Success in this course. The plan should include a weekly study schedule and a general list of things you think you will need to do each week to get the work done.

<br>


## 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

<img src="./Images/Columbus_taking_possession_of_the_new_country.jpg">

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).

<div class="QuestionsHeading">Answer the following questions:</div>
<div class="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.

<a href="javascript:showhide('Q1')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q1" style="display:none;">
* 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%.)

</div>
</br>

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

<a href="javascript:showhide('Q2')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q2" style="display:none;">

* There is only about a 2% chance that Columbus will draw out the marked Chick Pea.  This is not very likely.  
</div>
&nbsp;
</div>
</br>

**A Second Drawing**<br>
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).

<div class="QuestionsHeading">Answer the following questions:</div>
<div class="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?

<a href="javascript:showhide('Q3')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q3" style="display:none;">
* 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%.)
</div>
</br>

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

<a href="javascript:showhide('Q4')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q4" style="display:none;">
* It is very likely that Columbus would not draw out the marked chick pea.  This result is not surprising.
</div>
</br>

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?

<a href="javascript:showhide('Q5')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q5" style="display:none;">
* The sum of the probabilities is 1:

$$
\frac{1}{41} + \frac{40}{41} = \frac{41}{41} = 1
$$

</div>
&nbsp;
</div>
</br>

**Additional Drawings**<br>

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. <a href="javascript:showhide('bonusprobabilitycalc')"><span style="font-size:8pt;">(Show/Hide Solution)</span></a>

<div id="bonusprobabilitycalc" style="display:none;padding-left:50px;padding-right:50px;border:solid 1pt;border-color:orange;color:gray;">
**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.**

</div>


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.

<div class="QuestionsHeading">Answer the following questions:</div>
<div class="Questions">
6. If a fair, six-sided die* is rolled, what is the probability of rolling a 6?

<a href="javascript:showhide('Q6')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q6" style="display:none;">
* 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).
</div>
</br>

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

<a href="javascript:showhide('Q7')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q7" style="display:none;">
* 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."
</div>
</br>

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?

<a href="javascript:showhide('Q8')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q8" style="display:none;">
* 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.
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</br>

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?

<a href="javascript:showhide('Q9')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q9" style="display:none;">
* 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.

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$*$*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.*
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</br>

### 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.

<div class="QuestionsHeading">Answer the following questions:</div>
<div class="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?

<a href="javascript:showhide('Q10')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q10" style="display:none;">
* 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. 
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</br>

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?

<a href="javascript:showhide('Q11')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q11" style="display:none;">
* 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.
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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?

<a href="javascript:showhide('Q12')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q12" style="display:none;">
This can be viewed as one of two problems:  
</br> 

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%).

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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.

<div class="Emphasis">
&nbsp;&nbsp;**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.

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

&nbsp;&nbsp;You may have noticed that the Complement Rule is just a combination of the first two rules. 
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<div class="QuestionsHeading">Answer the following questions:</div>
<div class="Questions">
13. Which of the probability rules was violated by the statement in Question 10?

<a href="javascript:showhide('Q13')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q13" style="display:none;">
* Rule 1
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14. Which of the probability rules was violated by the statement in Question 11?

<a href="javascript:showhide('Q14')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q14" style="display:none;">
* Rule 1
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15. Which of the probability rules was violated by the statement in Question 12?

<a href="javascript:showhide('Q15')"><span style="font-size:8pt;">Show/Hide Solution</span></a>
<div id="Q15" style="display:none;">
* Rule 2 or Rule 3
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&nbsp;
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</br>

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.



<br>

## 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!

<br>

## Summary

<div class="SummaryHeading">Remember...</div>
<div class="Summary">

- 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 will be based on the readings, so study it well.
- Most weeks will cover two lessons
- By doing the work, staying on schedule, and living the Honor Code you *will* succeed in this class!

- 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. 
$$P(\text{not}~X) = 1 - P(X)$$

</div>
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## Navigation

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| **This Reading**               | **Next Reading**                                                       |
| :----------------------------: | :--------------------------------------------------------------------: |
| Lesson 01: <br> Course Introduction | [Lesson 2: <br> The Statistical Process & Design of Studies](Lesson02.html) |
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