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 "theorem - proof" format, we develop the material in an intuitive -- but still rigorous and mathematically precise -- 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 course are essentially the same as those of the corresponding MIT class (Probabilistic Systems Analysis and Applied Probability) -- a course that has been offered and continuously refined over more than 50 years. It is a challenging class, but it will enable you to apply the tools of probability theory to real-world applications or your research.

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Dolly 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 makes me realize how important it is to review according to the forgetting curve. The course support is excellent too!

When you struggle with problem sets that "threatens" to fill your weekends,say this to yourself:

"Doing things we know how to do well is enjoyable, and that’s exactly the opposite of what…

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 makes me realize how important it is to review according to the forgetting curve. The course support is excellent too!

When you struggle with problem sets that "threatens" to fill your weekends,say this to yourself:

"Doing things we know how to do well is enjoyable, and that’s exactly the opposite of what deliberate practice demands. Instead of doing what we’re good at, we should insistently seek out what we’re not good at. Then we identify the painful, difficult activities that will make us better and do those things over and over. If the activities that lead to greatness were easy and fun, then everyone would do them and they would not distinguish the best from the rest. The reality that deliberate practice is hard can even be seen as good news. It means that most people won’t do it. So your willingness to do it will distinguish you all the more." – by Geoff Colvin from "Talent is Overrated"

Personal story

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In China,math is the ticket to a good life.Most Chinese children from middle class background (actually, even Chinese parents who struggle to make ends meet would put their child's education above everything) have at least been exposed to Olympiad-level math by the age of 10 since it gives them a significant edge in the city-wide middle-school admission. I opted out of the program at the age of 11 after two years of understanding every word but not the whole sentence emerged out of the thin mouth of the teacher with a sphinxlike smile at the specialized weekend school for math and staying up to 1 am with my grandma (who also struggled to solve 5th grade Olympiad math problems despite graduating from medical school and generally considered intelligent) to finish assignments when everybody else seemed to do well in the class and had a normal childhood.

I need to point out that despite the tragic aura of my story, according to some of my friends,the specialized math program are the best part about childhood;it made them feel special or that they got where they are because of the foundational experience.

In China,contrary to the US,to be cool and popular,you need to excel in math and science.Your life depends on it.I can't think of one single guy whom a lot of girls have crushes is not a certified academic superstar.Caveat:From zero to one,look is still critical.

It was also the year that I started thinking about finding my comparative advantage in a zero-sum game (school admission) teeming with math geniuses who swallow math books with the same voracity I read fictions.I haven't found out yet.Talk about inequality.

I went to an okish middle school and worked so hard in the first two years that I have not been able to approach the same intensity. I neglected personal hygiene to memorize one more English word or solve one more physics problem.During our mandatory morning run around the playground (800 meters) ,I took my Chinese textbook with me so I could squeeze another poem into my hippocampus.I used torchlight during compulsory school talent show to read. I threatened to cut myself when my mom forced me to go to nonacademic events.Unsurprisingly,I was routinely the top ten in a class of 550 people.Thinking back,I was driven by fear that I would become mediocre like the rest of my peers.Chinese education system is engineered in such as way that a student is encouraged to link her self-worth with academic achievement.It could cause tremendous distress to those who have less than stellar grades.

After getting in the best high school in a city of 10 million people,I got 71/100 on the first physics midterm,which was way below the class average of 80.I remember asking the student who got 100 about how he did it.He said that he simply studied two weeks in advance and did all the problems in all the available practice books(like 4).Remember that our course load for tenth grade is 9 classes.Even if he was not smarter than me (at least in physics) before he did all the practice problems,he definitely was smarter after putting 10 more hours each week on the subject for 3 months.Now he's a physics major in a prestigious university in China.

Physics and maths,except at the Olympiad-level,is solely a matter of will and hard work. There is no talent to speak of, since the materials are presented in such a way that a person with a normal brain has enough processing power to understand and apply.

Enough personal background. My point is that even though I had certified intellectual disability at the age of 11*,it helped me build up a high tolerance for the very real physical pains of making incremental progresses on grasping abstract concepts or solving hard problems.

In an age of increasing distraction and automation, the only way to stay competitive is to work with the problems that you want to bang your head against the wall.That's the right level of intellectual challenge that will rewire your brain and make you a better problem solver.

*In the ultra-competitive Chinese system,not being exceptionally talented and hardworking could be considered a handicap since you are not gonna get in the best middle school and so on;the best universities in China has admission rate like 0.001 to 1 percent depending on whether you are from Beijing or not since universities differ only in selectiveness;the hardest to get in is the best.

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Gregorycompleted this course and found the course difficulty to be hard.

6.041x: Introduction to Probability - The Science of Uncertainty is a comprehensive 16-week 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 with differential and integral calculus to understand some of the material, and the course can easily take 10-15 hours per week. Given its 16-week duration, the time commitment required to get through everything is much higher than the average MOOC. The course touches on all the major topics you need to gain a solid understanding of probability i…

6.041x: Introduction to Probability - The Science of Uncertainty is a comprehensive 16-week 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 with differential and integral calculus to understand some of the material, and the course can easily take 10-15 hours per week. Given its 16-week duration, the time commitment required to get through everything is much higher than the average MOOC. The course touches on all the major topics you need to gain a solid understanding of probability including basic axioms of probability, conditional probability and independence, discrete and continuous random variables, Bayesian inference and the probabilistic underpinnings of classical statistics. The course grade is based on lecture comprehension questions, weekly homework assignments, 2 midterms and 1 final exam. The midterms are worth 15% apiece and the final is worth 30% so good performance on the exams is paramount to getting a good score. You need a total of 60% to pass and it isn't quite as easy to achieve that mark as it is in most MOOCs.

Weekly content consists of 2-4 lecture sequences covering different aspects of a particular topic in probability. Each lecture sequence contains about an hour of video in 5 to 15 minute segments and most video segments are followed by graded comprehension questions. The lecture videos themselves are crisp and the professor is good at explaining the material at a pace that doesn't overload you with too much information too quickly. There can be quite a bit of mathematical notation on the screen at times, but it is well-organized. Each week also has a series of solved problem videos where TAs walk you through applying the material in lecture to problems that are similar to those you will see in the homework. The solved problems sections add another 1 to 2 hours of video content per week.

Pure math courses usually aren't that fun because they spend a lot of time dealing with proofs and theory and not so much time dealing with the real world. This course can be a slog at times because it is long and there is a lot to absorb and remember, but after building up the basic tools of probably in the first few weeks, later weeks focus on more interesting extensions and applications. You won’t find another intro to probability with greater depth and breadth. This course is best suited for technical and math-minded people who will have to work with and apply probability in future coursework or in their professional lives. If you're looking for an intro that just gets you up to speed on the rudiments of every-day probability like coin flipping and dice rolling this course is overkill.

6.041x: Introduction to Probability is a great course for those serious about forming a solid foundation in probability. As professor Tsitsiklis states early on, "the first step in fighting an enemy like randomness is to study and understand your enemy." At the end of this course you will be armed with the tools necessary to wage a well-reasoned war against uncertainty.

I give 6.041x: Introduction to Probability 5 out of 5 stars: Excellent.

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 of proving and not proving concepts using math.

The best aspect about this course are the exercises. They are a valuable learning tool in themselves. Most exercises use sub answers and the problem often progresses to a more complex problem yielding in depth understanding of the material.

Many exercises c…

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 of proving and not proving concepts using math.

The best aspect about this course are the exercises. They are a valuable learning tool in themselves. Most exercises use sub answers and the problem often progresses to a more complex problem yielding in depth understanding of the material.

Many exercises can only be submitted once or twice which keeps me alert while learning and applying the knowledge. I consider this a good thing, but not everyone agrees on that. The Professor and TA's do an excellent job replying in the exercise forums where needed with the focus on increasing understanding.

It is a difficult course. Most people put in more than 12 hours per week. The fora show both grateful students because of the rigor and depth of the course and on the other side of the spectrum complaints about the amount work, it being too much for people with a full time job and family.

It starts of easy, but workload and complexity increase over time.

I highly recommended this one, but make sure you allot some time to keep up.

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Hchancompleted 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 exercise-driven course similar to what you'd get in-person 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.

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Siddharthcompleted 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.

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Jitingcompleted 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!

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Jinqiangcompleted 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 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 : )

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 an ordinary MOOC. Rigour is at the same level as you might expect in an excellent graduate level course taken on campus.

Having experienced this course, I can confidently say that advanced degrees can be offered fully online at the highest level of academic excellence. There is only one aspect that I feel unsatiated about and is the constraint of not having some one-on-one time with an instructor or TA and white board a concept that one may want to understand truly deeply.

Federico 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.

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.