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In Basic High School mathematics, you'll come across three approaches to probability:
Empirical Probability
Statistical Probability (Classical Probability)
Axiomatic Approach to probability
This course teaches the axiomatic approach to probability by discussing the theory first and then using many useful typical example. In the end, you will be able to calculate the probability of almost any typical event, as long as it is not beyond the scope of this text.
Read more
In Basic High School mathematics, you'll come across three approaches to probability:
Empirical Probability
Statistical Probability (Classical Probability)
Axiomatic Approach to probability
This course teaches the axiomatic approach to probability by discussing the theory first and then using many useful typical example. In the end, you will be able to calculate the probability of almost any typical event, as long as it is not beyond the scope of this text.
Read more
In Basic High School mathematics, you'll come across three approaches to probability:
Empirical Probability
Statistical Probability (Classical Probability)
Axiomatic Approach to probability
This course teaches the axiomatic approach to probability by discussing the theory first and then using many useful typical example. In the end, you will be able to calculate the probability of almost any typical event, as long as it is not beyond the scope of this text.
This course is also a part of a road map that takes from the basics of mathematics (pre-algebra - class 6) all the way up to calculus (class 12) based on the Indian system of education (NCERT). The text used here will give you a sturdy and robust foundation in mathematics provided that the whole road map is studied from beginning to end. The road map is highly recommended if you are planning a career in science, mathematics or engineering.
To access the road map, please search for "Great IT Courses" on the internet. on the website, please read the page titled as, "Mathematics 6-12 Standard". The same page has also been linked from the second lecture of the course.
To locate this particular course in the road map, please go to the page related to "Class 11". This course is title as, "16. Probability".
Thank you.
Register for this course and see more details by visiting:
OpenCourser.com/course/9nbiaj/axiomatic
What's inside
Learning objectives
- Axiomatic approach to probability
- Definition of random experiments, outcomes, sample space, events,
- Algebra of events
- Calculating the probability of an event using the axiomatic approach to probability
- Solving problems related to axiomatic approach to probability
- Learning how to use concepts and reasoning to reason you way through problems logically rather than memorization
- Axiomatic approach to probability
- Definition of random experiments, outcomes, sample space, events,
- Algebra of events
- Calculating the probability of an event using the axiomatic approach to probability
- Solving problems related to axiomatic approach to probability
- Learning how to use concepts and reasoning to reason you way through problems logically rather than memorization
Syllabus
Introduction
Instructor Introduction
Course and Road Map Info
In this Section, you will learn all the required definitions like Random Experiments, Outcome, Events, etc.
Read more
Syllabus
Introduction
Instructor Introduction
Course and Road Map Info
In this Section, you will learn all the required definitions like Random Experiments, Outcome, Events, etc.
Read more
In this video, we will talk about the properties of empirical probability and theoretical probability and why it's necessary to learn the axiomatic approach to probability.
In this video, we will define the terms mentioned in the video title and also discuss a couple of examples related to them.
In this video, we will define events based on a typical experiment.
In this video, we will talk about the occurrence of an event. We'll also talk about Impossible events, Sure events, Simple events and compound events that are different types of events.
In this video, we will talk about complementary events. We'll also talk about some common concepts related to sets such as intersection, union, etc. that will be used in the next videos.
In this video, we'll do a quick review of the concept of functions and relations since it's going to be used in one the upcoming definitions.
In this video, we will use an example to show how to calculate the probability of an event using the axiomatic approach to probability.
Example 12 #1
Example 12 #2
Example 12 #3
Example 13
Example 13 #2
Exercise 1 and 2
Exercise 3 #1
Exercise 3 #2, Exercise 4
Exercises 5, 6, 7
Exercises 7, 8, 9, 10, 11, 12
Exercises 13, 14, 15, 16, 17, 18
Exercises 19 and 20
Exercise 21
Example 14 #1
Example 14 #2
Example 15 #1
Example 15 #2
Example 16
Example 17
Exercise 1
Exercise 2
Exercise 3 and 5
Exercise 6 and 7
Exercise 8 and 9
Exercise 10
Traffic lights
Traffic lights
Read about what's good
what should give you pause and
possible dealbreakers
Provides a sturdy and robust foundation in mathematics, especially for those planning a career in science, mathematics, or engineering, by covering the axiomatic approach to probability
Focuses on developing logical reasoning skills, which are essential for problem-solving in mathematics and related disciplines, rather than relying solely on memorization
Is part of a larger roadmap spanning mathematics from pre-algebra to calculus, suggesting a comprehensive and structured approach to learning mathematics
Requires learners to locate the course within a broader curriculum roadmap, which may present a barrier to some learners who prefer self-contained learning experiences
Relies on the Indian system of education (NCERT), which may not align perfectly with the curricula of other educational systems, potentially requiring additional adaptation for some learners
Emphasizes the axiomatic approach to probability, which is a theoretical perspective that may not be as immediately applicable as empirical or statistical approaches for some learners
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Reviews summary
Foundational axiomatic probability
According to learners, this course provides a solid foundation in axiomatic probability, particularly praising the clear explanations and helpful examples that simplify complex ideas. Many found it perfect for learning from scratch and building confidence. While the content covers the syllabus well and provides useful exercises, some reviewers noted that the pace can feel slow at times and suggested the delivery could be more engaging. One learner found it difficult to follow, indicating it might not suit everyone or that some prerequisites are assumed, although the majority opinion leans strongly positive for beginners.
Provides a strong base for the topic.
"Great foundation building."
"A solid introduction. The explanations are good..."
"Perfect for learning the axiomatic approach from scratch."
"Covers the syllabus well. Provides a strong base."
Examples aid understanding of concepts.
"...The examples are very clear and help solidify the concepts. Great foundation building."
"Fantastic course! Explains complex ideas simply. The examples are well-chosen and practical."
"Very clear and concise. The examples are the best part, really helps connect the theory to practice."
"Some examples are a bit repetitive. Good exercises though."
Course content is explained clearly.
"Excellent course covering the axiomatic approach. The examples are very clear and help solidify the concepts."
"Perfect for learning the axiomatic approach from scratch. Definitions are clear, examples are helpful."
"Very clear and concise. The examples are the best part, really helps connect the theory to practice. Highly recommended..."
"Explains complex ideas simply."
Some learners found it hard to follow.
"Found it difficult to follow. The explanations weren't always clear to me, especially for the examples. Needed external resources."
Pace sometimes slow, delivery could be better.
"A solid introduction. The explanations are good, though sometimes the pace feels a bit slow."
"Content is okay, but the delivery could be more engaging."
"...the presentation is basic. Could be more dynamic. It's suitable for absolute beginners but might be too slow otherwise."
Activities
Be better prepared
before
your course. Deepen your understanding
during
and
after
it. Supplement your coursework and achieve mastery of the topics covered
in Axiomatic Probability - Mathematics with these
activities:
Review Set Theory Fundamentals
Show steps
Solidify your understanding of set theory, as it provides the foundation for understanding events and sample spaces in probability.
Browse courses on
Set Theory
Show steps
-
Review definitions of sets, subsets, and power sets.
-
Practice set operations like union, intersection, and complement.
-
Solve problems involving Venn diagrams and set notation.
Review 'Introduction to Probability' by Dimitri P. Bertsekas and John N. Tsitsiklis
Show steps
Deepen your understanding of probability theory with a widely respected textbook that covers the axiomatic approach in detail.
View
Melania
on Amazon
Show steps
-
Read the chapters related to axiomatic probability and sample spaces.
-
Work through the examples and exercises in the book.
-
Compare the book's explanations with the course material.
Help Others in Online Forums
Show steps
Solidify your understanding by explaining probability concepts to others in online forums or study groups.
Show steps
-
Find online forums or study groups related to probability.
-
Answer questions and provide explanations to other students.
-
Seek clarification from instructors or peers when needed.
Three other activities
Expand to see all activities and additional details
Show all six activities
Create a Probability Cheat Sheet
Show steps
Consolidate your knowledge by creating a concise cheat sheet summarizing key definitions, axioms, and formulas related to probability.
Show steps
-
Review the course materials and identify key concepts.
-
Organize the information into a clear and concise format.
-
Include definitions, axioms, and formulas.
-
Add examples to illustrate the concepts.
Solve Probability Problems from Past Exams
Show steps
Reinforce your problem-solving skills by working through a variety of probability problems from past exams or problem sets.
Show steps
-
Find a collection of probability problems from past exams.
-
Attempt to solve each problem independently.
-
Review the solutions and identify areas for improvement.
Review 'Probability and Random Processes' by Geoffrey Grimmett and David Stirzaker
Show steps
Expand your knowledge with a more advanced text that delves into the mathematical underpinnings of probability and random processes.
View
Probability and Random Processes: Fourth Edition
on Amazon
Show steps
-
Focus on chapters related to axiomatic probability and measure theory.
-
Work through the proofs and derivations in the book.
-
Compare the book's approach with the course's approach.
Review Set Theory Fundamentals
Show steps
Solidify your understanding of set theory, as it provides the foundation for understanding events and sample spaces in probability.
Browse courses on
Set Theory
Show steps
Show steps
Show steps
- Review definitions of sets, subsets, and power sets.
- Practice set operations like union, intersection, and complement.
- Solve problems involving Venn diagrams and set notation.
Review 'Introduction to Probability' by Dimitri P. Bertsekas and John N. Tsitsiklis
Show steps
Deepen your understanding of probability theory with a widely respected textbook that covers the axiomatic approach in detail.
View
Melania
on Amazon
Show steps
Show steps
Show steps
- Read the chapters related to axiomatic probability and sample spaces.
- Work through the examples and exercises in the book.
- Compare the book's explanations with the course material.
Help Others in Online Forums
Show steps
Solidify your understanding by explaining probability concepts to others in online forums or study groups.
Show steps
Show steps
Show steps
- Find online forums or study groups related to probability.
- Answer questions and provide explanations to other students.
- Seek clarification from instructors or peers when needed.
Three other activities
Expand to see all activities and additional details
Show all six activities
Create a Probability Cheat Sheet
Show steps
Consolidate your knowledge by creating a concise cheat sheet summarizing key definitions, axioms, and formulas related to probability.
Show steps
Show steps
Show steps
- Review the course materials and identify key concepts.
- Organize the information into a clear and concise format.
- Include definitions, axioms, and formulas.
- Add examples to illustrate the concepts.
Solve Probability Problems from Past Exams
Show steps
Reinforce your problem-solving skills by working through a variety of probability problems from past exams or problem sets.
Show steps
Show steps
Show steps
- Find a collection of probability problems from past exams.
- Attempt to solve each problem independently.
- Review the solutions and identify areas for improvement.
Review 'Probability and Random Processes' by Geoffrey Grimmett and David Stirzaker
Show steps
Expand your knowledge with a more advanced text that delves into the mathematical underpinnings of probability and random processes.
View
Probability and Random Processes: Fourth Edition
on Amazon
Show steps
Show steps
Show steps
- Focus on chapters related to axiomatic probability and measure theory.
- Work through the proofs and derivations in the book.
- Compare the book's approach with the course's approach.
Career center
Learners who complete Axiomatic Probability - Mathematics will develop knowledge and skills
that may be useful to these careers:
Financial Engineer
Financial Engineer
Financial engineers design and develop new financial instruments and strategies using mathematical and computational methods. Financial engineering requires a strong foundation in probability, statistics, optimization, and numerical analysis. This career path typically requires a master's degree or PhD. A course in axiomatic probability is directly relevant for a financial engineer, providing the theoretical basis for understanding and modeling financial risk. The course’s emphasis on random experiments, sample spaces, and calculating probability is useful for pricing derivatives, hedging risk, and developing trading algorithms. Understanding probability will lead to this career.
Quantitative Analyst
Quantitative Analyst
Quantitative analysts, often called quants, develop and implement mathematical and statistical models for pricing derivatives, managing risk, and identifying trading opportunities. These models rely heavily on probability theory, stochastic calculus, and numerical methods. A course in axiomatic probability is directly relevant for a quant, providing the foundational knowledge needed to understand and work with probabilistic models. The course’s focus on random experiments, sample space, and events helps build intuition and provides a theoretical basis for the complex quantitative models used in finance. This course is a starting point for anyone looking to become a quantitative analyst.
Actuary
Actuary
The role of an actuary involves assessing and managing financial risks, often within the insurance and finance industries. Actuaries analyze statistical data to estimate the probability and potential financial impact of future events. A course in axiomatic probability provides a strong foundation for this career, building skills in understanding random experiments, sample spaces, and events. The course’s focus on calculating probabilities using the axiomatic approach directly applies to the risk models actuaries develop, where they must use logic and reasoning. This course, with its coverage of events, sample space and calculating probability, directly prepares one to become an actuary.
Risk Manager
Risk Manager
A risk manager identifies, assesses, and mitigates risks that could impact an organization's financial stability, reputation, or operations. They develop risk management policies, implement controls, and monitor risk exposures. A course in axiomatic probability is useful for a risk manager, providing the theoretical basis for understanding and quantifying risk. The course’s coverage of events, outcomes, and probability calculations directly applies to assessing the likelihood and impact of various risks. This course prepares one for a career as a risk manager.
Statistician
Statistician
A statistician collects, analyzes, and interprets numerical data to draw conclusions and make informed decisions. Statisticians apply statistical methodologies to a wide range of fields, including healthcare, business, and government. A course in axiomatic probability is highly relevant for statisticians as it provides the theoretical underpinnings for understanding and applying probability concepts. The course’s emphasis on defining random experiments, outcomes, and events all help statisticians develop sound methods. This course, with its focus on the axiomatic approach to probability, helps statisticians develop robust foundations in mathematics.
Epidemiologist
Epidemiologist
Epidemiologists investigate the patterns and causes of disease and injury in populations. They use statistical methods, mathematical modeling, and public health principles to identify risk factors, track outbreaks, and develop prevention strategies. A course in axiomatic probability provides a foundation for understanding the probabilistic nature of disease transmission and the statistical methods used in epidemiological research. The course’s coverage of events, outcomes, and probability calculations directly applies to analyzing disease incidence and prevalence. The course helps an aspiring epidemiologist become more familiar with probability tools.
Operations Research Analyst
Operations Research Analyst
An operations research analyst uses mathematical and analytical methods to help organizations make better decisions about resource allocation, scheduling, and logistics. They develop models to optimize processes, improve efficiency, and reduce costs. A course in axiomatic probability is helpful for an operations research analyst, providing the basis for understanding and applying probability models in decision analysis. The course’s coverage of events, outcomes, and probability calculations is helpful for analyzing complex systems and optimizing resource allocation. The course helps to build background knowledge as an operations research analyst.
Data Scientist
Data Scientist
A data scientist uses statistical techniques, machine learning algorithms, and data visualization tools to extract insights from large datasets. Data scientists work with complex data to identify trends, patterns, and anomalies that can inform business strategies. A course in axiomatic probability is valuable for data scientists, as it provides the mathematical principles needed to understand probabilistic models and statistical inference. The course provides a foundation for understanding statistical probability, and gives a solid foundation for application of math in data science. Understanding of events, outcomes, and sample space provides a background for a career as data scientist.
Machine Learning Engineer
Machine Learning Engineer
A machine learning engineer develops and implements machine learning algorithms and systems. They work on building models that can learn from data and make predictions or decisions without being explicitly programmed. A course in axiomatic probability helps machine learning engineers understand the theoretical underpinnings of many machine learning algorithms, which rely on probabilistic models and statistical inference. The course is particularly helpful for understanding Bayesian methods and probabilistic graphical models commonly used in machine learning. The course, with its focus on axiomatic probability, can lead to a career as a machine learning engineer.
Research Scientist
Research Scientist
A research scientist designs and conducts experiments, analyzes data, and publishes findings in a specific field, such as biology, chemistry, or physics. This career often requires a doctorate. A course in axiomatic probability helps research scientists to analyze data and to draw statistically significant conclusions from experimental results. The course is especially helpful to those looking to work with stochastic models. The course's exercises and examples are helpful.
Financial Analyst
Financial Analyst
A financial analyst evaluates financial data, provides investment recommendations, and helps organizations make sound financial decisions. They analyze market trends, assess investment risks, and develop financial models. A course in axiomatic probability may be useful for a financial analyst, giving a starting point for evaluating risk and uncertainty in financial markets. The course’s exercises and examples in calculating probability of events helps financial analysts to apply their knowledge to a wide range of scenarios. The course's focus axiomatic approach to probability can provide a new prespective to a financial analyst.
Business Intelligence Analyst
Business Intelligence Analyst
A business intelligence analyst (BI analyst) analyzes data to identify trends and insights that can help businesses make better decisions. BI analysts use data visualization tools, statistical software, and database management systems to collect, clean, and present data in a meaningful way. A course in axiomatic probability may be useful for a BI analyst, providing a foundation for understanding statistical inference and hypothesis testing. The course’s coverage of events, outcomes, and probability calculations can help BI analysts draw more informed conclusions from their data analysis. The course helps one look at data in a new light.
Market Research Analyst
Market Research Analyst
A market research analyst studies consumer behavior and market trends to advise companies on product development, marketing strategies, and pricing. They collect and analyze data using surveys, focus groups, and statistical analysis to understand customer preferences and identify market opportunities. A course in axiomatic probability may be help a market research analyst better understand the probability when making decisions. The course's exercises help prepare for this career.
Software Engineer
Software Engineer
A software engineer designs, develops, and tests software applications and systems. They work on a variety of projects, from developing mobile apps to building complex enterprise software. A course in axiomatic probability may be useful for a software engineer, particularly those working on applications that involve probabilistic modeling or statistical analysis. The course's coverage of probability helps software engineers implement algorithms. This course can provide new foundations for a future software engineer.
Teacher
Teacher
Teachers are responsible for educating students in a variety of subjects. A mathematics teacher specializes in mathematics. Teachers at the high school and university level typically require an advanced degree. A course in axiomatic probability can provide a teacher with increased insight into probability. The additional insight will help students understand the material at a deeper level. As a result, a prospective teacher may find this course valuable. The course's exercises help to hone skills in the subject.
For more career information including salaries, visit:
OpenCourser.com/course/9nbiaj/axiomatic
Reading list
We've selected two books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Axiomatic Probability - Mathematics.
Offers a rigorous treatment of probability theory and random processes, suitable for advanced students. It provides a deeper understanding of the mathematical foundations of probability. While it may be more advanced than the course itself, it serves as an excellent resource for further study and exploration. It valuable reference for those pursuing careers in mathematics, statistics, or related fields.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/9nbiaj/axiomatic
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