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Bayes' Theorem Examples: A Visual Introduction For Beginners
From Google search results to Netflix recommendations and investment strategies, Bayes Theorem (also often called Bayes Rule or Bayes Formula) is used across countless industries to help calculate and assess probability.
Bayesian statistics is taught in most first-year statistics classes across the nation, but there is one major problem that many students (and others who are interested in the theorem) face. The theorem is not intuitive for most people, and understanding how it works can be a challenge, especially because it is often taught without visual aids.
In this guide, we unpack the various components of the theorem and provide a basic overview of how it works - and with illustrations to help. Three scenarios - the flu, breathalyzer tests, and peacekeeping - are used throughout the booklet to teach how problems involving Bayes Theorem can be approached and solved.
Over 60 hand-drawn visuals are included throughout to help you work through each problem as you learn by example. The illustrations are simple, hand-drawn, and in black and white.
For those interested, we have also included sections typically not found in other beginner guides to Bayes Rule.
These include:
A short tutorial on how to understand problem scenarios and find P(B), P(A), and P(B|A). For many people, knowing how to approach scenarios and break them apart can be daunting. In this booklet, we provide a quick step-by-step reference on how to confidently understand scenarios.
A few examples of how to think like a Bayesian in everyday life.
Bayes Rule might seem somewhat abstract, but it can be applied to many areas of life and help you make better decisions. It is a great tool that can help you with critical thinking, problem-solving, and dealing with the gray areas of life.
A concise history of Bayes Rule. Bayes Theorem has a fascinating 200+ year history, and we have summed it up for you in this booklet. From its discovery in the 1700’s to its being used to break the German’s Enigma Code during World War 2, its tale is quite phenomenal.
Fascinating real-life stories on how Bayes formula is used in everyday life.From search and rescue to spam filtering and driverless cars, Bayes is used in many areas of modern day life. We have summed up 3 examples for you and provided an example of how Bayes could be used.
An expanded definitions, notations, and proof section.
We have included an expanded definitions and notations sections at the end of the booklet. In this section we define core terms more concretely, and also cover additional terms you might be confused about.
A recommended readings section.
From The Theory That Would Not Die to a few other books, there are a number of recommendations we have for further reading. Take a look! If you are a visual learner and like to learn by example, this intuitive booklet might be a good fit for you. Bayesian statistics is an incredibly fascinating topic and likely touches your life every single day. It is a very important tool that is used in data analysis throughout a wide-range of industries - so take an easy dive into the theorem for yourself with a visual approach!
If you are looking for a short beginners guide packed with visual examples, this booklet is for you.
Bayesian statistics is taught in most first-year statistics classes across the nation, but there is one major problem that many students (and others who are interested in the theorem) face. The theorem is not intuitive for most people, and understanding how it works can be a challenge, especially because it is often taught without visual aids.
In this guide, we unpack the various components of the theorem and provide a basic overview of how it works - and with illustrations to help. Three scenarios - the flu, breathalyzer tests, and peacekeeping - are used throughout the booklet to teach how problems involving Bayes Theorem can be approached and solved.
Over 60 hand-drawn visuals are included throughout to help you work through each problem as you learn by example. The illustrations are simple, hand-drawn, and in black and white.
For those interested, we have also included sections typically not found in other beginner guides to Bayes Rule.
These include:
A short tutorial on how to understand problem scenarios and find P(B), P(A), and P(B|A). For many people, knowing how to approach scenarios and break them apart can be daunting. In this booklet, we provide a quick step-by-step reference on how to confidently understand scenarios.
A few examples of how to think like a Bayesian in everyday life.
Bayes Rule might seem somewhat abstract, but it can be applied to many areas of life and help you make better decisions. It is a great tool that can help you with critical thinking, problem-solving, and dealing with the gray areas of life.
A concise history of Bayes Rule. Bayes Theorem has a fascinating 200+ year history, and we have summed it up for you in this booklet. From its discovery in the 1700’s to its being used to break the German’s Enigma Code during World War 2, its tale is quite phenomenal.
Fascinating real-life stories on how Bayes formula is used in everyday life.From search and rescue to spam filtering and driverless cars, Bayes is used in many areas of modern day life. We have summed up 3 examples for you and provided an example of how Bayes could be used.
An expanded definitions, notations, and proof section.
We have included an expanded definitions and notations sections at the end of the booklet. In this section we define core terms more concretely, and also cover additional terms you might be confused about.
A recommended readings section.
From The Theory That Would Not Die to a few other books, there are a number of recommendations we have for further reading. Take a look! If you are a visual learner and like to learn by example, this intuitive booklet might be a good fit for you. Bayesian statistics is an incredibly fascinating topic and likely touches your life every single day. It is a very important tool that is used in data analysis throughout a wide-range of industries - so take an easy dive into the theorem for yourself with a visual approach!
If you are looking for a short beginners guide packed with visual examples, this booklet is for you.
197 pages, Kindle Edition
First published October 2, 2016
1217 people are currently reading
1151 people want to read
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Displaying 1 - 30 of 63 reviews
December 16, 2018
Think you want some help understanding better or getting into the Bayes Theorem. Two cases: you have a friend who can help over a coffee on some back of the envelope stuff or you buy this book.
This books ticks many boxes: Does one thing and does it well, explains with enough detail, small, cute, articulate.
Then why the 2 stars: One for its price it could have a little bit of more information, or examples. Two the dead tree / paper version is just rushed and should not be sold. It is a print out of probably Kindle's version, which means that there are MANY LINKS but with ZERO URLs. Text goes like "for XYZ see here" with here being underlined and the destination is nowhere to be found.
Summarizing: great idea, good concept, but although they could do something great, very poor execution on the printed version, for things that could be fixed in 2 hours.
This books ticks many boxes: Does one thing and does it well, explains with enough detail, small, cute, articulate.
Then why the 2 stars: One for its price it could have a little bit of more information, or examples. Two the dead tree / paper version is just rushed and should not be sold. It is a print out of probably Kindle's version, which means that there are MANY LINKS but with ZERO URLs. Text goes like "for XYZ see here" with here being underlined and the destination is nowhere to be found.
Summarizing: great idea, good concept, but although they could do something great, very poor execution on the printed version, for things that could be fixed in 2 hours.
September 17, 2021
Sách viết xoàng nhưng có lẽ bản thân định lý Bayes đủ thú vị để đọc rồi 👌 Top 1 công thức ai cũng nên biết.
October 25, 2016
A good book that covers the basics of Bayes Theorem. I like the alternative methods explained in the book as well as the explanations. I liked the book and think it was helpful.
December 25, 2016
Great Quick Intro
I appreciated this very appreciable treatment of the Bayes theorem for a beginner. The visual examples alongside the math helped cement the concept.
I appreciated this very appreciable treatment of the Bayes theorem for a beginner. The visual examples alongside the math helped cement the concept.
January 23, 2018
I could actually use Bayes’ Theorem to estimate for you in this review the odds of you enjoying the book, but I fear it’ll look quite arbitrary without the clear and helpful visual guides that the book provides, walking the reader through many examples in a step-by-step fashion, such that we get to not only know how to use the theorem in a functional fashion (say, plugging and playing with the numbers, as one might if learning it for a mathematics exam, practically learning the method by rote), but also to have a genuine understanding and overview of how it actually works and why.
Maybe, like me, you never got on with mathematics at school. In my schooldays, my problems in the subject were threefold: 1) a bad teacher 2) an active resistance to anything I didn’t find stimulating 3) a lack of awareness of how to work around dyscalculia. So instead, I challenged my teacher to prove things (he refused and/or was unable; this book in contrast contains a neat proof, by the way), I accused him of witchcraft when he produced correct numeric answers with no demonstration of how things worked, and I generally struggled with anything containing numbers.
Here instead, everything is presented in a clear and simple fashion—as the title suggests, largely visual—minimizing the need to juggle a lot of numbers and instead working chiefly with concepts, which I can grasp much more readily. Where numbers are necessary, they’re not onerous and they’re nothing whose calculations one couldn’t do on a phone if necessary.
In short, a clear and engaging primer in how Bayes’ Theorem works, how to use it, and how to rapidly estimate changes in probability so as to make better decisions.
If only books like this were used in schools, resulting in people better understanding stats and probability, the world might have a lot fewer problems than it does!
Maybe, like me, you never got on with mathematics at school. In my schooldays, my problems in the subject were threefold: 1) a bad teacher 2) an active resistance to anything I didn’t find stimulating 3) a lack of awareness of how to work around dyscalculia. So instead, I challenged my teacher to prove things (he refused and/or was unable; this book in contrast contains a neat proof, by the way), I accused him of witchcraft when he produced correct numeric answers with no demonstration of how things worked, and I generally struggled with anything containing numbers.
Here instead, everything is presented in a clear and simple fashion—as the title suggests, largely visual—minimizing the need to juggle a lot of numbers and instead working chiefly with concepts, which I can grasp much more readily. Where numbers are necessary, they’re not onerous and they’re nothing whose calculations one couldn’t do on a phone if necessary.
In short, a clear and engaging primer in how Bayes’ Theorem works, how to use it, and how to rapidly estimate changes in probability so as to make better decisions.
If only books like this were used in schools, resulting in people better understanding stats and probability, the world might have a lot fewer problems than it does!
October 29, 2018
Bayes Theorem has a serious power to describe reality through mathematical probability. I have written If you are interested in statistics and the power of Bayes Theorem, please do yourself a favor and read this book.
Here's an example:
Bayes Theorem is simple, but it's implications are tremendous. There are many statistical probabilities that at face value seem concerning, but when the information is properly plug-in, you see a different result.
Here's an example:
1. We know that the people with the flu have a headache and sore throat roughly 90% of the time.
2. We are aware the probability of having the flu, in general, is only 5%.
3. We know that 20% of the population in a given year will have a headache and sore throat at any given time.
P(A/B) = P(B/A) P(A)/ P(B)
P(Flu/Symptons)= P(Symptoms/Flu) P(Flu) / P(Symptoms)
P(A/B) = .9x .05 /.2 = .225 or 22.5%
After plugging in the numbers into Bayes Theorem we can conclude from this that if you have a sore throat and headache, you only have a 22.5% probability of having the flu.
Khan Academy provides an introduction! Check it out!
Here's an example:
Bayes Theorem is simple, but it's implications are tremendous. There are many statistical probabilities that at face value seem concerning, but when the information is properly plug-in, you see a different result.
Here's an example:
1. We know that the people with the flu have a headache and sore throat roughly 90% of the time.
2. We are aware the probability of having the flu, in general, is only 5%.
3. We know that 20% of the population in a given year will have a headache and sore throat at any given time.
P(A/B) = P(B/A) P(A)/ P(B)
P(Flu/Symptons)= P(Symptoms/Flu) P(Flu) / P(Symptoms)
P(A/B) = .9x .05 /.2 = .225 or 22.5%
After plugging in the numbers into Bayes Theorem we can conclude from this that if you have a sore throat and headache, you only have a 22.5% probability of having the flu.
Khan Academy provides an introduction! Check it out!
January 9, 2018
Bayes' Theorem Examples: A Visual Introduction for Beginners by Dan Morris makes this seemingly complex theorem more understandable. From the beginning of the book, the language of the book is such that the novice can begin to understand and comprehend the subject matter. Morris lays it out thus : 4 Ways that the Theorem is explained, 'Bayes' Theorem helps us update a belief based on new evidence by creating a new belief, it helps us revise a probability when presented with new evidence, it helps us change our beliefs about a probability based on new evidence, and helps update a hypothesis based on new evidence'. With visual examples, such as Venn Diagrams, Decision trees, Letters(coin flips) and physical objects, the theorem can be more readily explained. Morris lays out the structure of the book to further facilitate understanding, breaking it down even more, finally with a list of recommended readings that delve even more into the theorem. Ultimately the theorem is about statistics and probabilities, and how these work within structures like Google and other business models. Highly recommend for beginners to understand this theorem.
May 12, 2017
"English below"
با توجه به تب این روزهای انتخابات، ذکر این نکته خالی از لطف نیست که پیش بینی های علمی رایج عمدتا بر اساس قضیه بیز انجام می گیرند. آشنایی اولیه با این قضیه می تواند برای همگان مفید باشد و باعث شود دیدگاههای واقع بینانه تری بر اساس شواهد و اطلاعات داشته باشیم
As an introductory book, it is a short and useful text. Examples and graphics are fine, although sometimes repetitive. However, I was expecting to see more insights on how to understand our problem is approachable by this theorem or how the results can be updated by receiving more information. In addition, for evaluating everyday problems, it is not obvious how the provided solutions are related to this theorem's calculations. Finally, the author has just copied and pasted some text, a reason the length of the book is more than what it could be.
با توجه به تب این روزهای انتخابات، ذکر این نکته خالی از لطف نیست که پیش بینی های علمی رایج عمدتا بر اساس قضیه بیز انجام می گیرند. آشنایی اولیه با این قضیه می تواند برای همگان مفید باشد و باعث شود دیدگاههای واقع بینانه تری بر اساس شواهد و اطلاعات داشته باشیم
As an introductory book, it is a short and useful text. Examples and graphics are fine, although sometimes repetitive. However, I was expecting to see more insights on how to understand our problem is approachable by this theorem or how the results can be updated by receiving more information. In addition, for evaluating everyday problems, it is not obvious how the provided solutions are related to this theorem's calculations. Finally, the author has just copied and pasted some text, a reason the length of the book is more than what it could be.
July 25, 2018
I needed to be taught math like this in school!
Excellent first contact with a long-debated but now indispensible nugget of mathematical ingeniousness. As a probability tool, Bayes' has been used everywhere from breaking the Enigma code machines to locating derelict sailors at sea to accurately predicting elections. Dan Morris gives very clear explanations and guides the reader through easy-to-understand examples, reviewed over and over under different lights. What little math I've learned has been decaying since I finished high school, almost three decades ago, yet I felt confident and excited to learn a tool that can intuitively be used to boost decisions and confidence to face the uncertainties of life.
Excellent first contact with a long-debated but now indispensible nugget of mathematical ingeniousness. As a probability tool, Bayes' has been used everywhere from breaking the Enigma code machines to locating derelict sailors at sea to accurately predicting elections. Dan Morris gives very clear explanations and guides the reader through easy-to-understand examples, reviewed over and over under different lights. What little math I've learned has been decaying since I finished high school, almost three decades ago, yet I felt confident and excited to learn a tool that can intuitively be used to boost decisions and confidence to face the uncertainties of life.
August 7, 2019
Easy and useful
A perfect book to not only learn about but to work with the theory. The simple formula is the basis of understanding things from how to asses medical tests to how to detect spam. Even if your knowledge of probability is minimal this book will teach you quite a lot about Bayesian thinking. You will also find out why it should really be called Laplace’s Theorem, but even so Laplace already has more than enough named in his honor, so I’m quite happy with the honor going to the Bayes.
A perfect book to not only learn about but to work with the theory. The simple formula is the basis of understanding things from how to asses medical tests to how to detect spam. Even if your knowledge of probability is minimal this book will teach you quite a lot about Bayesian thinking. You will also find out why it should really be called Laplace’s Theorem, but even so Laplace already has more than enough named in his honor, so I’m quite happy with the honor going to the Bayes.
July 19, 2018
A very quick but useful introduction
This is a thorough and enjoyable yet basic introduction to Bayes’ Theorem. Just Bayes’ Theorem. If you are looking for a 1 to 5 hour exploration of a single equation with references for further study, this is for you. If you are looking for anything more than that, this isn’t for you. Buy the electronic version for easy access to the many hyperlinks in the text.
This is a thorough and enjoyable yet basic introduction to Bayes’ Theorem. Just Bayes’ Theorem. If you are looking for a 1 to 5 hour exploration of a single equation with references for further study, this is for you. If you are looking for anything more than that, this isn’t for you. Buy the electronic version for easy access to the many hyperlinks in the text.
April 14, 2023
While reading "The Signal and the Noise" I encountered Bayes in a bit of hand waving fashion. Since it was critical to a real understanding I decided to stop and dig a little deeper. This book was the perfect fit. I followed the first couple problems then was able to solve them myself and whiz through the book. I dreaded but wanted to know the proof and it turned out to be simple and sort of obvious if looked at the right way.
January 2, 2018
Descripción totalmente introductoria del Teorema de Bayes. Solo se dan ejemplos resueltos (de una manera bastante sencilla y explicada) y aplicaciones a la vida real sin entrar en mucho detalle. Lo recomiendo solo a personas que no han tenido un acercamiento con la estadística porque podría llegar a aburrir a pesar de que es corto si sabes algo del tema.
January 12, 2018
Easy and fast Intro with real life (simplified) Examples
Recommended for anyone who wants to have some basic and quick understanding on Bayes. Quick and easy book. The author nicely points out the areas which can turn into difficulties when one starts using it for complex problems.
Recommended for anyone who wants to have some basic and quick understanding on Bayes. Quick and easy book. The author nicely points out the areas which can turn into difficulties when one starts using it for complex problems.
April 29, 2020
A very light introduction to a very profound topic.
This book is one of the best introductory texts for those who want to learn Bayes's theorem and it's applications. The best part is it gives an idea of the profundity of the subject by listing practical applications of the theorem, while keeping the narrative simple and light.
This book is one of the best introductory texts for those who want to learn Bayes's theorem and it's applications. The best part is it gives an idea of the profundity of the subject by listing practical applications of the theorem, while keeping the narrative simple and light.
August 21, 2025
A visual introduction for beginners. Neat and easy read for understanding Bayesian statistics. Good introduction and easy way to understand the formula. I'll need some more practice with these ideas. I liked how I felt mentally when trying to think about and solve the problems presented in the text.
July 18, 2017
Simple, straightforward
Nice introduction. Would have liked to learn how to deal with slightly more complex scenarios, but I feel ready to do that on my own either way, after reading this.
Nice introduction. Would have liked to learn how to deal with slightly more complex scenarios, but I feel ready to do that on my own either way, after reading this.
August 6, 2017
Okay as a fast and. Simpler overvaluation
This book is about as fast and simple as it gets. That dose not necessarily mean you will get it but this is a good place for the challenged to start.
This book is about as fast and simple as it gets. That dose not necessarily mean you will get it but this is a good place for the challenged to start.
August 28, 2017
I found this book well put together. Lots of incremental examples to help get your head around the theorem. I liked the idea of doing the same calculations using different methodologies. My only suggestion is that it could use some humor to lighten up the content.
October 31, 2017
Good, but very repetitive
The book has some good examples and illustrations, but they are too similar to each other. The visual "solution" by venn diagrams are definitely not useful the third time over. Recommend for people looking for good ways to explain Bayes to others.
The book has some good examples and illustrations, but they are too similar to each other. The visual "solution" by venn diagrams are definitely not useful the third time over. Recommend for people looking for good ways to explain Bayes to others.
June 20, 2018
Very basic - simple and easy to follow
If you want to learn Bayes Theorem more in-depth you will need to do more research and reading. The author warns you about this at the very beginning.
If you want to learn Bayes Theorem more in-depth you will need to do more research and reading. The author warns you about this at the very beginning.
June 23, 2018
A Great Introduction to Bates Theorem
This is a good place to start for anyone interested in understanding Bates Theorem. I recommend getting the Kindle Edition and reading it on a Fire or other device that allows you to review the internet links.
This is a good place to start for anyone interested in understanding Bates Theorem. I recommend getting the Kindle Edition and reading it on a Fire or other device that allows you to review the internet links.
January 17, 2019
This book provides a nice introduction to understanding and using Bayes’ Theorem to assess probabilities and make decisions. It provides easy to understand guidance that makes the process accessible to those who are not logicians or mathematicians.
September 8, 2019
It is written in the simplest manner to explain you the actual meaning of bayes theorem and how it works.
The examples are pretty realistic and you can correlate with them.
All in all, it's a good book for beginners!
The examples are pretty realistic and you can correlate with them.
All in all, it's a good book for beginners!
January 19, 2020
Gives the most basic theory in a digestible way. Only touches on the first couple of weeks of a course. It doesn't bother with updating models based on new information. Worth a read if you're new to the subject matter, but don't expect anything comprehensive.
July 6, 2017
An excellent and graphic (pun intended) introduction to WHAT the famous theorem is and how to use it.
August 7, 2017
Good foundation book for probability.
August 18, 2017
A good book with practical approach
A good introductory material with real life examples. Also contains links to interesting articles that enrich the content in this book.
A good introductory material with real life examples. Also contains links to interesting articles that enrich the content in this book.
August 25, 2017
A bit repetitive
An interesting book for beginners but examples seems repetitive to me with the same text and steps. Half of the book is similar to the other half.
However, images and descriptions are really nice to understand the concepts.
An interesting book for beginners but examples seems repetitive to me with the same text and steps. Half of the book is similar to the other half.
However, images and descriptions are really nice to understand the concepts.
September 28, 2017
Very simple but functional beginner's introduction to the main concepts of Bayes Theorem. I thought the presentation was a little lazy though, with lots of copy-paste of descriptions and links to external articles instead of trying to incorporate summaries of said articles into the book.
Displaying 1 - 30 of 63 reviews