Probability  The Science of Uncertainty and Data
Massachusetts Institute of Technology via edX

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Overview
The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions.
Probabilistic models use the language of mathematics. But instead of relying on the traditional "theoremproof" format, we develop the material in an intuitive  but still rigorous and mathematicallyprecise  manner. Furthermore, while the applications are multiple and evident, we emphasize the basic concepts and methodologies that are universally applicable.
The course covers all of the basic probability concepts, including:
 multiple discrete or continuous random variables, expectations, and conditional distributions
 laws of large numbers
 the main tools of Bayesian inference methods
 an introduction to random processes (Poisson processes and Markov chains)
The contents of this courseare heavily based upon the corresponding MIT class  Introduction to Probability  a course that has been offered and continuously refined over more than 50 years. It is a challenging class but will enable you to apply the tools of probability theory to realworld applications or to your research.
This course is part of theMITx MicroMasters Program in Statistics and Data Science. Master the skills needed to be an informed and effective practitioner of data science. You will complete this course and three others from MITx, at a similar pace and level of rigor as an oncampus course at MIT, and then take a virtuallyproctored exam to earn your MicroMasters, an academic credential that will demonstrate your proficiency in data science or accelerate your path towards an MIT PhD or a Master's at other universities. To learn more about this program, please visit https://micromasters.mit.edu/ds/.
Syllabus
Unit 1: Probability models and axioms
 Probability models and axioms
 Mathematical background: Sets; sequences, limits, and series; (un)countable sets.
Unit 2: Conditioning and independence
 Conditioning and Bayes' rule
 Independence
Unit 3: Counting
 Counting
Unit 4: Discrete random variables
 Probability mass functions and expectations
 Variance; Conditioning on an event; Multiple random variables
 Conditioning on a random variable; Independence of random variables
Unit 5: Continuous random variables
 Probability density functions
 Conditioning on an event; Multiple random variables
 Conditioning on a random variable; Independence; Bayes' rule
Unit 6: Further topics on random variables
 Derived distributions
 Sums of independent random variables; Covariance and correlation
 Conditional expectation and variance revisited; Sum of a random number of independent random variables
Unit 7: Bayesian inference
 Introduction to Bayesian inference
 Linear models with normal noise
 Least mean squares (LMS) estimation
 Linear least mean squares (LLMS) estimation
Unit 8: Limit theorems and classical statistics
 Inequalities, convergence, and the Weak Law of Large Numbers
 The Central Limit Theorem (CLT)
 An introduction to classical statistics
Unit 9: Bernoulli and Poisson processes
 The Bernoulli process
 The Poisson process
 More on the Poisson process
Unit 10 (Optional): Markov chains
 Finitestate Markov chains
 Steadystate behavior of Markov chains
 Absorption probabilities and expected time to absorption
Taught by
Patrick Jaillet and John Tsitsiklis
Charts
 #2 in Subjects / Mathematics / Statistics & Probability
 #3 in Subjects / Data Science
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Reviews
4.9 rating, based on 32 reviews

Dolly Ye is taking this course right now, spending 14 hours a week on it and found the course difficulty to be hard.
This course is very thorough and challenging.The learning curve could be steep at times,but don't get frustrated. The problem sets are structured to deepen your understanding of the materials.I partially completed it a year before,but taking it again... 
Gregory J Hamel ( Life Is Study) completed this course and found the course difficulty to be hard.
6.041x: Introduction to Probability  The Science of Uncertainty is a comprehensive 16week introduction to probability offered by MIT through the edX MOOC platform. Although this course is dubbed an “introduction” it is not easy. You need familiarity... 
Bart completed this course.
The best courses I have taken. I have some prior experience but that is very rusty and definitely not as extensive as the topics in this course. The professor has a very clear way of explaining the topics. There is hardly any repetition and a good balance... 
HChan completed this course, spending 8 hours a week on it and found the course difficulty to be medium.
Many online courses are watered down in some way, but this one feels like a proper rigorous exercisedriven course similar to what you'd get inperson at a top school like MIT. The professors present concepts in lectures that have obviously been honed to a laser focus through years of pedogogical experience  there is not a single wasted second in the presentations and they go exactly at the right pace and detail for you to understand the concepts. The exercises will make you work for your knowledge and are critical for really internalizing the concepts. This is the best online course I have taken in any subject. 
Siddharth Gupta completed this course, spending 14 hours a week on it and found the course difficulty to be medium.
This is the most beautifully designed course I have ever attended in my life, having completed first 4 weeks, it has been both rigorous and to the point. This course questions the whole pedagogy that I have faced in India. If someone fails to attempt the questions after the lectures, it is because of his inefficiency of grasping the concepts, the questions are designed in such a way that they would test your learning to the core. 
Anonymous is taking this course right now.
I am taking a graduate version of the course numbered MITx 6.431 to end at the end of Dec'18. It is the best MOOC I have ever taken. In fact, given what is generally thought of MOOC in terms of rigour, and level of difficulty, this course is nothing like... 
Jiting Tian completed this course, spending 16 hours a week on it and found the course difficulty to be hard.
This is an introductory course on probability theory, but, it's very hard (after all, it's from MIT). The materials, which have covered all the related topics on probability, are organized quite well and illustrated in a gradual and clear way. A lot of difficult exercises are required, but they are very useful to help students understand the concepts and master the calculation ways. The whole course lasts for 16 weeks (oh my god!), but when I insist on to the end, I have learnt so much and feel so satisfied. Thank you, Prof. John Tsitsiklis and the course staff! 
Jinqiang Zhang completed this course, spending 10 hours a week on it and found the course difficulty to be very hard.
I tried 2 or 3 times for the course, it's very hard. It's hard because it has more content than a usual probability course. The professor is very good, nice accent, smart guy.
It's good to have some calculus knowledge prior to this course, because you don't want to handle the difficulty from the course itself, as well as the technique issue from calculus. At the end of the course, the classical part of the probability is quite different from the front parts, I don't feel I had a firm grasp of the ideas, I guess at some point I need to revisit this part. 
Soumyadeep Roy is taking this course right now, spending 8 hours a week on it and found the course difficulty to be medium.
This is my 2nd online course from MIT. It's indeed the best introduction to probability theory I've ever had. I had no intuition about the subject,and moreover I used to think it's something which can't be done by myself. But as the course is going on, I'm finding myself not only good in probability,and it has also created a love for probabilistic models that ,I guess,truly govern everything around us. Enjoying so far : ) 
Arnaud Dion is taking this course right now, spending 4 hours a week on it and found the course difficulty to be medium.
This is a great introducing course on probability. A certain level in math is a prerequisite, but nothing complicated. The teacher is clear and the his explanations really help to understand notion that can appear complicated at first glance. The exercices are designed to help the understanding. They're not "challenging", but are helpful. 
Federico Carrone is taking this course right now, spending 12 hours a week on it and found the course difficulty to be hard.
This is a great introducing course on probability. A certain level in math is a prerequisite, but nothing complicated. The teacher is clear and the his explanations really help to understand notion that can appear complicated at first glance. The exercices are designed to help the understanding. They're not "challenging", but are helpful.

Anonymous is taking this course right now.
This course is just perfect! One of the best moocs you can find ever! It covers a lot, and it's rigorous and demanding. But everything is explained very clearly and the course team help a lot. Thanks for the team and hope there would be more courses coming from the team. 
Prashant SIngh completed this course, spending 40 hours a week on it and found the course difficulty to be hard.
Its a very good course which deals with concepts.
Quizzes tests your understanding.
Requires good books for help.
Requires effort.
High quality video .
High quality quizzes.
But I feel some concepts could be explained easily. 
In my opinion this course is one of the top5 best courses on MOOC. Its is not a washout course for mass consumption or cheap way to receive a certificate, but it is really challenging and requires a lot of time to follow the lectures, read the textbook,...

Luiz Cunha completed this course, spending 8 hours a week on it and found the course difficulty to be hard.
Maybe my number 1 MOOC taken. This is top quality content. Highly recommended
Everything is good in this MOOC.
And it is kept being updated 
Harish Ramakrishnan completed this course, spending 9 hours a week on it and found the course difficulty to be hard.
One of the best ever MOOCs. Very challenging. This should be in top 10 MOOCs of all time. Overall it is worth all the time we invest 
What to say? It is simply THE best MOOC you can find, given that you have the time to study the subject.
You don't need neither the interest nor a good knowledge on the subject: the interest will just come by attending the course teach by these superb professors, previous knowledge is NOT needed, as everything is introduced very gently.
Still, you will "arrive" to have strong intuitions and solid mathematical foundations to the subject.
A course not to miss! 
Anonymous is taking this course right now.
So happy about this course so far. I'm currently on the 4th unit out of 7.
The style of this course is to start from complete basics and then slowly building complexity. This is often what courses try to do, but they rarely succeed this well in my opinion. I feel like I have a solid grasp on the topics that I've completed thus far.
Thanks to EdX for providing this amazing free resource! 
Federico Carrone is taking this course right now, spending 12 hours a week on it and found the course difficulty to be hard.
It is an introductory course but, I would not recommend it to somebody that doesn't have any idea of probability or statistics. The problems and exams have some difficult exercises. After taking many probability courses, this is the first time I feel I really understand it. It is fun and rigorous at the same. It is one of the best MOOC I've taken if not the best. 
Anonymous completed this course.
This course is really a legend! It's very challenging but professor explained all the things so well and smoothly. I have taken a course in probability and found it so hard that I cannot understand a lot of theory. But through this course, I really understand a lot.