Introduction

In this course, we will discuss the definition of conditional probability and its application to solving real-world problems. Probability is a central concept in mathematics used to quantify the likelihood of uncertain events occurring. It is used extensively in the UVic courses MATH 151 and STAT 252. The concept of probability originated in the 17th century and evolved with the study of gambling problems (Literature Cited).
In addition, probability has important applications in fields such as science, engineering, and business. For example, with conditional probability, we can analyze the outcome of rolling dice or drawing playing cards to make data-driven decisions.

Learning Outcomes

• Understand the definition and apply the formula of conditional probability.

• Distinguish between dependent and independent events

• Describe at least one industry’s Application of Conditional Probability in Real-Life Situations.

Read and Watch

https://www.youtube.com/watch?v=KzfWUEJjG18 (12 min) – Basic Probability

https://www.mathsisfun.com/data/probability.html (15min) – Demonstrates the concept of the probability basic rules

https://doi.org/10.1007/978-1-4612-4374-8 – (20 min) The use of probability

Concept

Definition of Conditional Probability

Definition of Conditional Probability: Conditional probability describes the probability of another event occurring when an event is known to have happened. The formula is as follows:

Where:

• P(A∣B) is the probability of event A occurring under the condition that event B occurs.
• P(A∩B) is the probability that event A and event B occur simultaneously.
• P(B) is the probability that event B occurs.

Example:

Prerequisite: There are 60 students. 30 students take math, 20 students take physics, and 15 students take both subjects.

Question: Given that a student takes physics, what is the probability that they also take math?

Solution: 

Dependent & Independent Events

Dependent Events: Two events are dependent if the occurrence of one affects the probability of the other.

Formula:

Example of a Dependent Event：

Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.

Formulas:

Example of an Independent Event：

Word version

Venn Diagrams：

We can also use Venn diagrams to distinguish between Dependent events (intersecting) and Independent events (non-intersecting).

Part III: Practical Applications and Importance of Conditional Probability in Real Life

Conditional probability is a very important concept in mathematics and it has many wide applications in real life. Through conditional probability, we can analyze data and future directions based on in known conditions. I will explain the application of conditional probability in mainstream scenarios in order to help you better understand the application of conditional probability in life.

Medical field
First of all conditional probability plays a very important role in disease detection and diagnosis.
For example, there is a disease test with a positive rate of 95%, but only 1% of people are actually infected with the disease. By using conditional probability, we can calculate the probability that people will actually get the disease if they test positive, which helps doctors to make a diagnosis and reduce misdiagnosis.

Risk management
Second, insurance and financial companies can use probability to help them assess risk.
For example, insurance companies can calculate the likelihood that a customer will get sick based on their health status and lifestyle habits. This can help them calculate the cost of insurance for different groups of people.

Machine learning
Conditional probability is one of the foundations of recommendation algorithms and classification models.
For example, in our commonly used e-commerce platforms predict the items that users may like based on their search history and purchase history before displaying them on the customer’s homepage. E-commerce platform based on the condition that the user has already purchased a certain product, and then go to calculate the probability of purchasing other related products.

Assessment

Learning Activity

Have students cut two large circles out of colored cardstock:

• The blue circle represents the 80 households that have pets
• The yellow circle represents the 50 households that have gardens

Draw a large box on the board to represent 120 households and have students come up to the table and place the two circles:

• The overlapping part of the two circles is 30 households (who have both pets and gardens)
• Ask students to observe and say what they see

Ask questions to guide:

• “Of the 80 households that have pets, how many also have gardens?”
• “So what is the probability that among the households that have pets, there is a garden?”

Formula Correspondence:

• Write the conditional probability formula: P(B|A) = P(A∩B)/P(A)
• A is “having a pet” and B is “having a garden”.

Identify the media you created and the tools you used.

For the video portion of this assignment, I used an iPad to record the video and imovie to edit the video. The video part is mainly responsible for explaining and analyzing the example problems to help students understand and master the knowledge more intuitively. At the same time, students can watch the video anytime and anywhere, and adjust the speed to help them learn.

Finally, I installed the H5P plugin in wordpress to create practice questions for the students. The type of questions I chose were multiple choice and added detailed comments and hints to each option to help students read and understand the questions. With these questions, I can test the students’ mastery of the basic rules, and the difficulty of the questions will keep increasing to help them consolidate their learning. The main reason I chose H5P is that it provides an interactive element, such as the multiple choice questions we used in this case. This interactive element can increase students’ engagement and motivation to learn and improve learning outcomes. At the same time, students can get instant feedback and correct answers to help them deepen their impressions.

Identify the principles, theories and techniques

Firstly, we have used the principle of coherence in the videos to ensure that there is no background music in the videos, only the human part, to ensure that other voices do not interfere with the students’ ability to watch the videos. In addition, I followed Voice, I always used real human voices in the videos instead of choosing to use AI-generated audio files to help students feel more immersed. Also, I adhered to the Segmenting principle in the production of the videos by making sure that each video was no longer than five minutes to ensure that the students would not be distracted by the length of the video, which would lead to a decrease in efficiency. Lastly, I used the Personalization Principle in the videos, and the overall feeling of the videos was relaxed and enthusiastic without giving the viewer an overly serious atmosphere. I also used the Personalization Principle in the video.

In addition to this, I also used the Image Principle in my Venn diagram design so that the students could understand the theorem of probability better. Also, I used Personalization Principle, at one minute and 05 seconds in the second video, I posed a question to the audience, ‘did you notice something’. It creates a relaxing and enjoyable learning atmosphere for the students and they will be curious and active to think about the question. These two parts make the students attracted to my course and ENGAGE in the course.

Lastly, I used a back design throughout the course with the content being distributed into big ideas, Learning Outcomes, Evidence of Learning, Assessments, and Learning Activities. The reason I chose the back design is because I was using the back design because I was talking about the math aspect of the course and I wanted to make sure that it was not too long. I am talking about math, and back design helps me to give my students clear learning objectives to start with, and then effectively plan instruction and assessments based on their math knowledge.

Determine what media you included from other sources and why you chose it

Youtube: Math Antics – Basic Probability

This video provides students with basic definitions and simple usage of concepts, which can help students learn the content of the course better and can be used by students as a pre-course study.

Textbook: Probability

This textbook is a very professional mathematical knowledge textbook, the explanation and ideas are systematic, structured and logical, can be used as a comprehensive guide to the concepts, the book has a large number of examples with answers and exercises to facilitate the application of knowledge.

Website: Probability formulas

This site provides a comprehensive compilation of probability formulas and sample problem analysis. Each probability theorem and formula includes clear explanations and examples so that readers can easily grasp the concepts. In addition, it contains the vast majority of conceptual formulas, and learners can quickly find the resources they need on the site. Finally, the resource is also helpful for students taking a probability course, as it provides a handy reference for basic concepts and problem-solving. For example, it provides calculations for common probability distributions and statistics.

Lesson plan

Big Idea:

• Conditional probability is an important tool for understanding the relationship between events. It helps us make reasonable predictions and decisions in the face of uncertainty.

Course Objectives:

• understand the definition and formula of conditional probability.
• distinguish between dependent and independent events
• 描述条件概率在现实生活中的应用

Learning outcomes:

• learn how to apply conditional probability to solve real-world problems (e.g., dice rolling, card drawing problems).
• distinguish between Dependent Events & Independent Events
• Describe at least one industry where probability is applied, explain how it is used in each, and explain the possible outcomes or benefits of the application.

Evidence of learning:

• correctly calculate conditional probability
• participate in exercises to apply what they have learned to solve real-life problems

Assessments:

• Two sets of in-class problems (one each on rolling dice and drawing cards) on conditional probability.

Learning activities:

Have students cut two large circles out of colored cardstock:

• The blue circle represents the 80 households that have pets
• The yellow circle represents the 50 households that have gardens

Draw a large box on the board to represent 120 households and have students come up to the table and place the two circles:

• The overlapping part of the two circles is 30 households (who have both pets and gardens)
• Ask students to observe and say what they see

Ask questions to guide:

• “Of the 80 households that have pets, how many also have gardens?”
• “So what is the probability that among the households that have pets, there is a garden?”

Formula Correspondence:

• Write the conditional probability formula: P(B|A) = P(A∩B)/P(A)
• A is “having a pet” and B is “having a garden”.

Reference:

DeMichele, Thomas. “Probability Theory Was Invented to Solve a Gambling Problem.” Fact or Myth?, factmyth.com, factmyth.com/factoids/probability-theory-was-invented-to-solve-a-gambling-problem. Accessed 29 Nov. 2024.

“History of Probability.” International Journal of Science and Research, www.ijsr.net/get_abstract.php?paper_id=ART20203456. Accessed 29 Nov. 2024.

Khan Academy. “Bayes Theorem.” YouTube, uploaded by Khan Academy, 20 Jan. 2021, https://www.youtube.com/watch?v=KzfWUEJjG18.

Pitman, Jim. Probability. Springer Science & Business Media, 2012.

“Probability.” Math is Fun, 2023, https://www.mathsisfun.com/data/probability.html. Accessed 30 Nov. 2024.

What learning experience does a branched narrative like Twine provide for learners?

I think branching narratives like twine give learners the freedom to explore in the direction they want, which can greatly increase their desire to explore. Because they don’t just have one choice, they can try the next path if they fail in one ending, and each option has a different story behind it. Learners will look forward to different story endings, try different options, and increase engagement in reading. In addition, learners can more easily step into the shoes of the characters and imagine that they are the protagonists of the story. For example, when I read a branching narrative I’ve written, I imagine myself as the prince rescuing the princess who has been robbed by the dragon.

Where else have you encountered branching narratives?

In a video game I once played, ‘Divinity: Original Sin’, a game famous for its branching narratives. Every decision the player makes, any action they take affects the direction of the game. For example, the relationship between the character and the gang, whether the character is a villain or a hero. As I played, I was immersed in the game and had a great sense of immersion. I don’t stop playing once and keep trying different endings and story lines.

Which of the 7 storytelling techniques used by the most inspiring TED speakers did you find most compelling? What techniques did you recognize in their talks?

I think the speaker in this article, Richard Turere, gave the most compelling performance. Firstly he used visual technique, in his presentation he showed the audience photographs and some simple text through which he showed the audience how his lighting system protects his livestock from lions and did not use complicated text or background. The audience was able to visualize the scene in their minds and understand how the lighting system worked by using the photos provided by Richard Turere. Secondly, he uses the technique of telling a personal story, a story from his own experience, which allows the audience to become more immersed.

In addition, Richard Turere provides hand-drawn drawings so that the audience can clearly enter his world and immerse themselves in the story. Even though the drawings are not very elaborate and are somewhat generic, they greatly help the reader to understand the principles and effects of lighting and to become more immersed in the story he is telling. Finally, Richard Turere’s use of photographs does not distract the reader but rather serves as a supplement to help the reader become immersed in the story and visualize the scene. Also, the story ends on a positive note as Richard Turere successfully uses his lighting system to protect his livestock.

Describe a meaningful learning experience that began with a story you heard. How did it affect you?

The Pursuit of Happyness, which is a very well known inspirational movie. Before I watched this movie, I had heard my parents tell me this story and it had a big impact on my life. The story is about the hero who through constant hard work gets a formal finance job and eventually starts his own company, going from a poor man to a rich man through his own hard work. On his way to success he encountered many difficulties, such as divorcing his wife, having no place to live and no money to buy food. But he never gave up and believed that he would be successful. Finally, he overcame the difficulties and showed himself bravely, and finally realized his dream.

The experience of the protagonist in the story gave me great inspiration and made me believe that as long as I keep working hard towards my dreams and have strong beliefs that I will succeed, there will be a happy ending in the end. During my freshman year of college, I returned to my home country due to an epidemic. Due to the 12-hour time difference between China and North America, I was often sleep deprived and could not listen to the lectures attentively, and my learning efficiency was very low. I failed the midterm exams of many courses, and I once wanted to give up and continue studying in Canada. Because of this story, I started to try different ways to make up for my lack of knowledge and increase my study efficiency, such as watching videos and self-study, and eventually I got good marks at the end of the semester.

What senses did it inspire?

This story are used visual senses. As my parents told the story, they described the environment of the main character living in an outhouse with his son: dark and damp. I could imagine what kind of harsh environment the protagonist was in and his hardships. In addition, I could feel the excitement and joy in the protagonist’s heart when he succeeded in being officially hired by the company.

Storyboard

Video

Comments

What did you find when you ran the WAVE accessibility report on your blog post?

After downloading the WAZE plugin in my browser and applying it to my moudle 1 blog, I realized that my blog was missing significant contrast in many areas, which resulted in readers not being able to quickly get information when viewing my site. For example, I didn’t take a different type of font size or bold in areas that I thought were more important. This causes the reader to see the same enlarged font for both body text and keywords, which can be misleading and prevent them from doing a quick read. I also did not add any titles or notes above my video, which can confuse readers as to what the video is about and why it exists.

Also, through the WAZE report I found out that I didn’t optimize for people with visual impairments. I need to make sure that my font colors and sizes are easy for them to read and differentiate between, for example, if I don’t distinguish between the title and the body of the video, they won’t be able to tell what part they are reading. At the same time, I needed to avoid typographical errors that would make it impossible for visually impaired people to find the layout. Finally, I needed to include headings for each section to help guide the reader where to read.

Text-to-speech tools

I often use text-to-speech on social media programs where I ask Siri to read out the message for me. For example, when I’m driving and don’t have access to my phone, or don’t have my hands free to control my phone. In addition, when I don’t want to look at the phone screen but want to read a book, I will use text-to-speech to read the book and let the AI read the article. When I use text-to-speech tools, I have the same idea as Mayer’s Voice Principle, although with other programs like Siri they try to simulate the human voice and add emotion to the reading. I’m still not able to concentrate as much as I would if I were listening to a real person, because I can clearly feel that there’s a machine voice in there that isn’t really emotional, which makes it impossible for me to immerse myself in it. For example, the voice of Siri makes me feel more cold-blooded, a machine reporting back to me on what I’m doing.

What does inclusive design mean to you?

Inclusive design is a very important presence for me. inclusive design is not just for the visually impaired, it is for all people. It includes religion, language, race, culture. For example, I am not a native English speaker or a Canadian. When social media platforms send me abbreviated slang or complex phrases, I don’t understand them, and even though I’ve studied and speak English, I don’t understand all of them. With Inclusive Design, trying to use simple words and phrases can significantly in helping people with poor reading skills to quickly understand the meaning without getting lost.

What design principles are used in posters designed with Canva？

I used the Coherence Principle in my poster, where I didn’t use useless pictures or other decorative elements. Instead, I simply drew a table to categorize the content so that readers can easily understand and read it. At the same time, I made sure that all the font colors are uniformly black, and white background. This makes it easy to read for both general and visually impaired readers. I also made sure that the fonts were formatted consistently in Times New Roman, and I used the Signalling Principle to add bold fonts and colors to the subheadings in the table so that readers could quickly find the parts they wanted to read, and I also made the lines and symbols for each of the knowledge points so that they were clearer to read.

Comments

What has been your experience with video game learning support? What principles seem to be widely used in in-game support?

I’ve encountered a lot of Temporal Contiguity Principles in games, where text and images appear at the same time, allowing students to make connections between the two. One of the most classic uses of this principle in games is to play an animation at the beginning of the game with a narrator to help the player understand the context of the game. For example, war, alien invasion. In addition, the Pre-training Principle is a very common principle used in games. For example, before the game starts, the game will have newbie exercises on how to move, attack, and dodge. At the same time, the game gives the player a series of options that allow the game system to understand the player’s level of play and generate a tutorial. Lastly, all the games I’ve played have adhered to the Voice Principle, which is that voice actors’ voices are more emotive and realistic than artificially generated voices, and they make for a more immersive gaming experience for the player.

Almost all the games I’ve played don’t adhere to the Coherence Principle, which requires that unimportant content not be included, and favors the use of images and text to communicate content efficiently, and no background music. However, games often choose to use animation to communicate content rather than images, and are accompanied by background music to provide immersion in the game.

Where do you see constructive adjustment and reverse design being used in this course or other courses you are taking/have taken?

In the math I have taken the professor used Constructive Alignment in the design of the course. Constructive Alignment requires the course to follow a sequence of learning objectives, activities, and assessments. In MATH 151 the professor would explain the end goal of the class in the first class of the semester, which is what we would end up with when we finished the class. The professor in MATH 151 will explain the final goal of the class in the first class, which is what we will know when we finish the class. The professor will then set up learning activities such as weekly math lab exercises and quizzes, and set up group assignments for students to work on projects to help reinforce the knowledge gained in the class. Finally, the professor began to develop a grading rubric that would determine the percentage of points and the distribution of grades for each knowledge point.

Backward design

Backward design starts with the objectives before the assessment and learning activities. In STAT 252, which I have taken before, the professor first set up learning objectives for us, which required proficiency in calculating probability by the time the course was completed. Then the assessment program was set up where the instructor would determine if we could pass the course based on the results of each exam. Finally, learning activities were set up where we were required to complete assignments from the professor’s step-by-step program and form groups to work together on projects.

How will you use multimedia tools and content to support and enhance active learning? What will it allow them to do that they aren’t doing now?

I would try to use multimedia tools to design something that has leaderboards, fun learning, interactive learning, and promotes students’ desire to keep learning new things. An example of this is a popular program for learning English called Duolingguo. The software has a community to help students share their experiences with each other, and it distributes the learning content into different modules and uses games to help students memorize new words. Kahoot, a program I used to use in middle school and high school, allows teachers to design different types of questions, multiple choice and fill-in-the-blanks. Students can get higher scores by answering them quickly, so they can keep thinking faster and memorize faster.

What does it allow them to do that they aren’t doing now?

From my personal experience, I believe that using VR or MR can do things that educational tools have not been able to do before. VR and MR allow students to learn without being confined to a specific environment or condition, for example, students can use VR equipment to literally walk into a historical site, or enter the world of Haiti to observe topographical features and various types of fish. Instead of just being located in a picture in a book, students can observe with their eyes, making learning more interactive and fun.

Can you describe (step-by-step) an example of scaffolding in a learning experience you have designed or experienced?

The professor in STAT 252, which I studied last year, used scaffolding to help us learn.

First the professor started the course by going over the basic concepts of statistics and gave us the basic use of other tools such as excel. Examples of statistics are median, mean, and any use of excel for summing, multiplying.

The professor will then explain example problems and break them down into many smaller parts to show us how to answer them. For example, breaking down a problem into calculating statistics for different data sets and the variance in them.

After that, the professor gives the students practice problems, making sure that they are not too difficult and are similar to the examples. The professor gradually reduces the number of hints given to the students to ensure that they are able to complete the problems independently.

In addition, the professor encourages group discussions among students, so that students can share their problems with each other to solve them, rather than relying on parsing to get the answers directly. At the same time, the students will exchange their work with each other for correction, and find out the deficiencies of other students to explain and correct them.

Finally, the professor will provide a detailed evaluation after completing all the questions and help students to find out the areas that need to be strengthened. Students also interact with the professor to find out what they need to fix to make sure that they can get better at future assignments.

How was your experience with H5P?

It was my first time to hear about H5P, and I tried many different types of H5P in the Catalogue of H5P Content. One of the things that impressed me the most was the use of H5P to learn Japanese. I need to keep filling in the correct Japanese words in the video to keep the video playing normally, and if I input the wrong words, they will provide me some hints, which can increase my learning concentration and efficiency. At the same time, the designer can specialize some key knowledge points and set independent questions for them. This can help students to become proficient in a part of the lesson that they don’t know and need to review before moving on to the next step. Also, I have created some simple computational topics using h5p for this assignment. And I added hit to each option to show students how to find the correct method. I think it is a very good tool to help learning.

What activities do you think you would utilize most often in an instructional setting and what would you use them for? Which ones do you think would require the most resources to create?

I would create some videos using H5P and insert some questions into the videos. Firstly I would make sure that the videos are interesting to ensure that the students would want to understand the development and plot behind them to increase their engagement with the content. Instead of just reading about the knowledge alone, which would bore the students. In addition, special questions will be created for some important points so that students can grasp the knowledge quickly. Lastly, I believe that videos require the most resources to create because they require editing scripts, finding actors, locations, and spending a lot of time filming and editing to get a great video.

Comments:

I agree with you about h5p, it is an excellent educational tool and I try to use it for making multiple choice questions and videos to help with learning. It can quickly help learners consolidate and review what they just learned in class and will keep them focused. In addition, I also think that the preparation of the course is the most time-consuming, for example, we need to go to prepare a variety of materials to create interactive types of projects, for example, video games, videos, multiple choice questions and other types of h5p projects.

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