Very good, clear and concise lecture from Prof. Tsitsiklis. The teachers at my university may be extremely skilled in their field of study, but unfortunately many of them do not have the ability to pass on the knowledge to us students in an eloquent manner... These MIT-lectures are very much appreciated and I have used, and will continue to use them when needed during my undergraduate as well as graduate years!
The same is here, I have watched these videos and scored well in my courses of Operation research, Regression Analysis, and Econometrics, now planning for Probability by these videos. Thanks a lot to MIT open courseware
I started watching this back in April as a college freshmen and it inspired me to study and take my first actuary exam. I passed this September. Thank you MIT
This is an absolutely brilliant introduction to the theory of Probability. Many high school textbooks define probability as a frequency of possible outcomes to the total outcomes and get into counting problems. Students would quickly lose interest while solving myriad counting problems involving cards, dies, etc. and never see the bigger picture of probability. Here the professor instead choose to give a broader theoretical context behind the probability in the first lecture. Think what he did. He clearly defined his objective as to give a math framework to uncertainty faced in any field and started to systematically define the terms and rules. To begin with, we could call any uncertain activity that we are interested in, as an experiment and in that experiment, we could define a relevant sample space (he gives examples of irrelevant sample space to stress the importance of defining relevant ss) and then a probability law which quantifies our believes about how the outcomes in the sample space might occur. He stresses on this fact that probability law can be anything and need not be an empirical frequency which we usually observe. Probability of heads in a toss need not be 1/2 or probability of getting any one of the value when a six-sided die is rolled need not be 1/6. We assume them to be 1/2 or 1/6 because we empirically observe it and also logically it makes sense to assign those values when you know that the coin or the dice is not at all biased in any way. But some dices or coins due to their deformities could favor certain outcomes and need not have equal fairness to all outcomes. This is such a revelation and not many realise this. He deals with uniform probability - finite sample space separately to make this point more clear. Also, the formulation of events that we are interested as subsets of possible outcomes and using pictures to find the probabilities of those events is absolutely beautiful. He lays down the fundamental axioms and uses them wonderfully to derive the intuitive idea that probability of an event or a subset is nothing but the sum of probabilities of outcomes which are part of that event or subset. When we look at an event this way and also deal with uniform probability law (prob of each outcome is 1/N), we get probability of event as n/N where the formulation comes from sum of n 1/Ns rather than dumbly solving counting problems for numerator containing possible outcomes of event and denominator containing all outcomes. This way of thinking helps one to understand and use probability theory beyond counting problems. Similarly, equivalence of probability to the areas while dealing with continuous distribution makes lot of sense and problem boils down to finding out the areas that our events occupy.
I used this course for passing my Exam P from SOA. Great great course! Cannot recommend it enough. Prof. Tsitsiklis has an amazing ability to make abstract concepts super intuitive. e.g. Never had to remember any formulae for discrete space conditional probability. Transforming into the sub-universe was enough to compute all probabilities. All the TAs are great as well. Big Thank you to Prof. Tsitsiklis and everyone involved and of course MIT OCW for making this available to everyone! I am heading straight to the donations page :-)
Prof. Tsitsiklis can not be thanked enough for his outstanding capability to bring immediate clearness into these topics. These videos truely are an invaluable contribution to education.
informative without excess fluff, intrinsically entertaining, and well paced. i hope the rest of the lecture series is as well done as this first introduction
This is the best probability course I have come across online. Checked out several courses but it has been simplified here that makes it very clear for us who graduated from school for a while. The teaching style is outstanding. Thank you MIT and Professor John Tsitsiklis
I'm attending to an statistics class at University of Buenos Aires because I'm following Economics, and I miraculously ran into this. Incredible lecture!!!
Yeah, these guys are the best at what they do. But there is no need to let down other teachers to praise them. You can't expect everyone on the world to be the best.
Being taught by an instructor who not only has an h-Index of 90 but is also the author of your textbook is a flex you can only have while sitting in an MIT classroom.
personal notes : when assigning probabilities to various parts of the sample space, we do not assign them to individual parts fo the sample space, rather to subsets of the sample space.
My professor of simulation and modelling (course that relies heavily on these concepts) in my university is a fan of this guy. Always quotes him and tells his stories. I am happy I found his lectures :)
Thank you so much, great teacher, I had to retake Probability subject 2 times, and never quite grasped it, teacher only followed what was written in the book, there were no added insights, with this single lecture I got it so much better.
Probability was a very difficult math course when I was in university which was made even harder by the professor who taught it. At least now, I can appreciate the subject more.
🎯 Key Takeaways for quick navigation: 00:00 ☕️ *The video is an MIT OpenCourseWare lecture on probability models and axioms.* 00:56 📚 *Lecturer John Tsitsiklis emphasizes the importance of the head TA, Uzoma, in managing the course logistics.* 02:23 🗂️ *Tutorials and problem-solving play a crucial role in understanding the subtleties and difficulties of probability.* 03:50 🔄 *The process of assigning students to recitation sections involves an initial assignment with a chance of dissatisfaction, allowing resubmission for adjustment.* 05:46 📖 *The class focuses on understanding basic concepts and tools of probability rather than memorizing formulas.* 07:40 🌐 *Probability theory provides a systematic framework for dealing with uncertainty, applicable across various fields.* 10:30 🎲 *Lecture aims to cover the setup of probabilistic models, including the sample space, probability law, and axioms of probability.* 11:25 📋 *Sample space for an experiment is a set of all possible outcomes, described as mutually exclusive and collectively exhaustive.* 16:12 🎲 *In a two-roll dice experiment, outcomes are properly distinguished, leading to a sample space of 16 distinct possibilities.* 19:58 🔍 *Distinguishing between results and outcomes in a sequential experiment helps clarify concepts, as seen in the dice example.* 21:23 🌐 *Sample spaces can be finite or infinite, illustrated with examples of a dice experiment and a dart-throwing experiment.* 21:52 🎲 *Probabilities are assigned to subsets of the sample space, called events, rather than individual outcomes. The probability of an event is the numerical representation of the belief in its likelihood.* 24:10 📏 *Probability values must be between 0 and 1, with 0 indicating certainty of non-occurrence, and 1 indicating certainty of occurrence.* 27:29 🧀 *The third axiom states that for disjoint events A and B, the probability of A or B occurring is the sum of their individual probabilities, resembling how cream cheese spreads over sets.* 29:55 🔄 *Probability values are derived to be less than or equal to 1 using the second axiom, the third axiom, and the fact that the probability of the entire sample space is 1.* 31:47 🔗 *The probability of the union of three disjoint sets is the sum of their individual probabilities, a property derived from the additivity axiom for two sets.* 34:10 🎲 *For finite sets, the total probability is the sum of the probabilities of individual elements, simplifying calculations.* 36:34 🤔 *Some very weird sets may not have probabilities assigned to them, but this is a theoretical concern and not relevant in practical applications.* 37:29 🎲 *Setting up a sample space, defining a probability law, and visualizing events allows for solving various probability problems systematically.* 42:37 📐 *Problems under the discrete uniform law, where all outcomes are equally likely, often reduce to simple counting, making calculations straightforward.* 44:00 🎯 *In continuous probability problems, like the dart problem, assigning probabilities based on the area of subsets of the sample space allows for solving problems using the same principles as in discrete cases.* 45:25 📐 *Visualizing events using a picture aids in understanding and calculating probabilities, as demonstrated in the example of finding the probability of the sum being less than 1/2.* 46:24 🧮 *Calculating probabilities involves using the probability law, where the probability of a set is equal to the area of that set, as demonstrated through examples.* 47:22 🔄 *The countable additivity axiom is introduced, allowing for the legitimate addition of probabilities of an infinite sequence of disjoint sets, addressing scenarios like flipping a coin until obtaining heads.* 49:42 🔍 *The countable additivity axiom is more robust than the previous additivity axiom, enabling the addition of probabilities for an ordered sequence of disjoint events, a crucial concept for handling infinite collections.* Made with HARPA AI
Thanks for this lesson, e-learning will be the future, we have all theory, advice books in the intro, we have all to study, this e-learning will be helpfull also because many people will be at home and this => minus traffic , so minus caos in the city.
HI guys am a student at BOTHO UNIVERSITY IN BOTSWANA STUDING COMPUTER SCIENCE.I realy like Proff John Tsitsiklis.PROBABILITY MODELS AND AXIOMS wish i attended at M.I.T .....
A couple of questions: 1) if a single element has 0 probability, why a singleton has a probability greater than 0? 2) the first additivity axiom and the countable additivity axiom both say that the probability of the union of disjoint events is equal to the probability of the sum of each individual probability. In what they actually differ?
1) Whether a single element, i.e. a singleton, has non-zero probability depends on the probability law. For discrete uniform distributions it will be always non-zero and for continuous probability distributions it will always be zero. 2) For the first additivity axiom, the union is the union of two sets and can be extended to any finite union by induction. For the countable additivity axiom, the union is a countable union, so this is a stronger axiom.
Let A and B be two events such that the occurrence of A implies occurrence of B, But notvice-versa, then the correct relation between P(A) and P(B) is? a)P(A) < P(B) b)P(B)≥P(A) c)P(A) = P(B) d)P(A)≥P(B) Correct answer of this question ? Please tell
If P(AUB) = P(A)+P(B) for independent events then why is the following solution wrong? Roll 4 4 sided dice. What are chances of getting at least on 4? P(AUBUCUD) = P(A)+P(B)+P(C)+P(D) = 1/4+1/4+1/4+1/4. I know this is wrong so save your breath in posting the correct solution which is 1-(3/4)^4. Why is the solution wrong? It's a good question and will raise an important point.
talking about splitting hairs, at 20:10 he states that the sample space is infinite. but he must be talking about impossibly small darts. all real objects are confined within the limits of plank space and plank increments. although the number is very large, there is actually a finite number of real spaces that a real dart could land on within that square. the plank units are what stops infinite regression within a limited area of space in actual real world applications. only in thought experiments can you have points that are smaller than plank units of size.
I guess we had better stop doing math if we can only work with things that don't only exist as thought experiments, ie. small and large numbers (as in extremely), n-dimensions, circles... I mean all of these things have amazing application to the real worth but don't and/or can't really exist in nature.
Yeah, that's why it's a thought experiment and he is not actually suggesting to do an experiment of throwing darts with an infinitely small tip at the real numbers. It is just a way to visualize the property of a real number: having a chance of 0 to be chosen at random out of an interval. What I find quite interesting is that the chance to chose any number with a finite description is zero as well.
I am curious if there's another statistics class on OCW that is recommended as well as this course on probability? This is the class I think will benefit me most but I think a statistics class with video lectures would be excellent.
At the near end of the lecture, his meaning of strengthening is the ability to consider the probability of the union of countably infinite disjoint events which is not possible by the earlier axiom.
A city records a population of 23,000 in 2006 The statistical agency projects that by 2011, the city will hit a population of 34,000 1. How can we calculate what the population may have been in 2007, 2008, 2009, and 2010 2. How can we calculate the percentage of increase in each of these years? 3. How can we estimate the population in 2012, 2013, 2014, 2015 and 2016? Thank you
If you assume that the population grows at a constant rate you can find out. Call a yearly growth multiplier x. When you multiply 23,000 by x five times over five years you get 34,000. 23x^5 = 34 x = (34/23)^(1/5) x is the fifth root of 34/23, or about 1.0813099921... The answer to 2. is x minus 1 converted to a percentage. You can use this method to go forward in time by multiplying the population by x, or backwards by dividing it by x.
Hello guys , i am going to be taking a class in probability and statistics this coming semester , if anyone has followed these videos , do they cover statics as well or they are biased on probability and they touch both subjects well
This is just probability. Probability theory is the foundation on which statistics is built. This course is good but will not teach you most of the things you will learn in a statistics course.
We presume you mean the difference between 18.650 and 6.041 (this video is for 6.041). 6.041 is lower level course and you are not required to know probability theory. 18.650 requires knowledge of probability theory. See the 6.041 course on MIT OpenCourseWare at ocw.mit.edu/6-041F10 for more information. Best wishes on your studies!
