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
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 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
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.
🎯 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
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.
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.
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.
The meat starts at 09:56 for those who don't want to waste their time. 21:50 I'm not sure if this is correct :p The probability of 0 assigned to each of these points would mean that there is _absolutely no chance_ I hit _any_ of them :P So I wouldn't be allowed to hit anywhere inside that square, and I'm also not allowed to hit outside of it, so I simply am not allowed to throw the dart at all :P What would be correct to say, I think, is that if the precision _approaches_ infinity, the probability _approaches_ 0, but it is not _exactly_ 0, just something arbitrarily close to it.
+Bon Bon In order to have a probability the sum of all possibilities must be a 1 (Some number must always occur) when the probability of hitting a single space becomes infinite and measuring all possible outcomes becomes impossible the sum of all possibilities =/=1 so the probability is 0. Look at it from a mathematical and not a physical science perspective it is just an infinite example.
Alex Buck _"In order to have a probability the sum of all possibilities must be a 1"_ Tell me something I _don't_ know... _"Some number must always occur"_ That's exactly my point, and my main objection to that particular moment of the lecture. Because in his example with points in a square, all points are supposed to have probability of 0, which would imply that _none_ of them can occur, and yet, there _must_ occur at least _one_ of them, since the entire square has probability of 1. That's a *contradiction* in plain sight, something a *mathematician* should be aware of after all. _"when the probability of hitting a single space becomes infinite"_ That would be wrong too, because probability cannot exceed 1. So it cannot be infinite either. But it can be *infinitesimal* (_very close_ to 0, but _not yet_ 0). Infinitesimal is not the same as infinite. _"Look at it from a mathematical and not a physical science perspective"_ That's exactly the perspective I am looking at it. _"it is just an infinite example"_ Infinite or not, it cannot and shouldn't lead to contradictions. See also my objections to his next lecture. There's more of it.
+Bon Bon Infinity in itself is a paradox if treated as a real number. The area of that square is not defined therefore can be treated as infinite and a point that WAS defined would have a probability of 0 as the area of the square is infinite. A point by itself has no area as defined by geometry and therefore when testing for it's occurrence on an area must =0.
Alex Buck Infinity is a paradox only when treated as a _number_ (a number in general, not just a real number), which is incorrect. That's why it _shouldn't_ be treated as a number, and it is an error to do so. There is, however, nothing paradoxical in infinity itself. It is just an idea that tells you that you can continue some process without bounds. That's how all the ancient philosophers treated it: as a _potential_, not as something you actually need to walk through. And the cause of the paradoxes in this probability problem is not because of the infinities (infinitely many points in a square), but because of treating points as if they had areas, which contradicts their definition: Points have no dimensions, so they don't have any area. (Have you seen my comment under his other lecture yet? I explain it there in more detail.) _"The area of that square is not defined"_ What? :| What do you mean it's "not defined"? It's a 1x1 square, Celestia dammit! It's area is 1. _"therefore can be treated as infinite"_ First of all: _non sequitur_. Second: no, it cannot be treated as infinite. It is 1, a finite number. It is the _number of points_ on the square which is infinite, but this has nothing to do with areas at all. _"A point by itself has no area as defined by geometry"_ Exactly. _"and therefore when testing for it's occurrence on an area must =0."_ Not necessarily. Having no area doesn't necessarily mean area = 0. Thoughts have no area as well, but it doesn't mean that they have area of 0, because they are not geometrical objects either. The problem with this approach lays not in infinities, but in the assumption that the square is _made of_ points (infinitely many), and therefore that the area of the entire square (1 in this case) is a sum of the areas of points (which are assumed to be 0). Too many incorrect assumptions, and it always leads to paradoxes. The original assumption was that probabilities in that square are measured by areas (the area of the entire square being the primary example: equal to 1). So talking about the areas of the individual points on that square is nonsense, because it contradicts that assumption. We are supposed to measure probabilities with areas, not with points, right? We can subdivide the square into smaller and smaller areas, but they will _never_ be points. They will always be some smaller areas (of polygons or some other geometric figures). They can shrink around some point only _in a limit_, but this is not the same as _becoming_ a point (approaching is not the same as becoming). Therefore, infinitesimal areas of smaller and smaller shapes are OK, 0 areas of points are not OK.
There are no paradoxes. Learn something about calculus and you will see it's pretty exact and rigid science. In the context of this science, infinity exists, it's well defined, and what the professor said is true...
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.
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.
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.
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
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.
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.
+mohammad abdullah no you getting that wrong, it's a diagonal intersecting the x and y axis in (1/2,0) and (0,1/2)respectively, (.25,.25) is actually a point on that diagonal
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.
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.
Congratulations to those who are watching this in 2024. Wishing you best of learning. 🎉
Index for you guys
5:46 Introduction to the course
9:50 beginning of first class
25:50 third axiom
42:15 discrete uniform law example
thanks :)
Thanks
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!
I thought its only case in India !!! 😯🙀
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
Lies again? Lecture Room
@@cyanide4u539 what videos did you use for Econometrics ?
@@sohamshinde1258 are you doing ECO HONS ? will this be beneficial if your ans is yes
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.
everything makes more sense when passed through a set theory lens :)
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
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 :-)
Did you complete any other exams since?
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.
🎯 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
This is amazing!
This is overwhelming that u guys r providing lectures for free to the needed ones.
Great job MIT.
is this in sync with stats for ECONOMICS HONS at delhi university ?
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
"...perhaps we're splitting hairs here, but perhaps it's useful to keep the concepts right." I have always wanted to have a Professor like him.
All these lectures are GOLD. Thank you MIT for making them available for free.
Hey MIT, you guys rock for putting up such exquisite material online for free. I will definitely be making a small donation for such a great cause.
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!!!
Maarttiin denunciado lince
Verpiss Dich ya me estoy depsidiendo d emi cuenta
Maarttiin qué picardía
National University of Cordoba presente papah
i really don't know who still goes to school?:-)
we have everything on the internet :-)
thank you, mit guys!
Genius Professor and simple explanation
I can't breathe normally while listening to his lecture :)
Anyway Prof. John Tsitsiklis has really helped me clear those important concepts.
Raise your hand if you are still watching it in 2024
this lecturer is the best lecturer i've ever had. never encounterd such clear explanations! very recommende :-)
John Tsitsiklis is a God. He exemplifies what amazing teachers are. Grateful beyond words.
Jump to 10:00, that is where the fun begins.
Saving students from crappy Teachers since Nov 9, 2012... THANK YOU!!!
REPLY
Mrs. Riddell true that!
Absolutely.
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.
Raise your hand if you are watching it in 2023!
Okay. Can i put it down now?
@@raspian1019 😀, haha
Mee
2024
@@raspian1019 hahhaha yeah you can if hand is alive
What an elegant way of lecturing. Thank you, sir.
WOW....
Awesome ...Professor is direct, to the point, simple and comprehensive at the same time in explaining the concepts....
Great lecture Professor Tsitsiklis, very clear, pretty neat as well as the ones from your TA´s. I am following MIT OpenCourseWare.
This professor is so fabulous! One of the best professors ever!
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.
The best Probality Theory course I've seen.
GOAT Lectures as first course in probability
Nice prof. Wish I had this guy when I was struggling in my own Stats class years ago.
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.
Starts at 10 minutes
i watched for like 5 minutes before seeing this comment..lol
10:00 so you have something to click on
Eyvallah cigerim
Ty 😉
Thanks buddy
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.
Thanks for making probability easy for us...u r really a good teacher.
..no doubt.
This man is a genius at explaining.
Oh, this old-school projector is so nice)
yellow + blue = green , he teach art too!
“Think of probability as cream cheese…”. 😂 This was such a a helpful lecture, thank you!
This is amazing!
The lectures are really nice and very detailed!
(completed)
thank Sir....!
I guess that not all MIT lecturers are great but I am very sure this guys is really amazing. I like him
The meat starts at 09:56 for those who don't want to waste their time.
21:50 I'm not sure if this is correct :p The probability of 0 assigned to each of these points would mean that there is _absolutely no chance_ I hit _any_ of them :P So I wouldn't be allowed to hit anywhere inside that square, and I'm also not allowed to hit outside of it, so I simply am not allowed to throw the dart at all :P What would be correct to say, I think, is that if the precision _approaches_ infinity, the probability _approaches_ 0, but it is not _exactly_ 0, just something arbitrarily close to it.
+Bon Bon In order to have a probability the sum of all possibilities must be a 1 (Some number must always occur) when the probability of hitting a single space becomes infinite and measuring all possible outcomes becomes impossible the sum of all possibilities =/=1 so the probability is 0. Look at it from a mathematical and not a physical science perspective it is just an infinite example.
Alex Buck _"In order to have a probability the sum of all possibilities must be a 1"_
Tell me something I _don't_ know...
_"Some number must always occur"_
That's exactly my point, and my main objection to that particular moment of the lecture. Because in his example with points in a square, all points are supposed to have probability of 0, which would imply that _none_ of them can occur, and yet, there _must_ occur at least _one_ of them, since the entire square has probability of 1. That's a *contradiction* in plain sight, something a *mathematician* should be aware of after all.
_"when the probability of hitting a single space becomes infinite"_
That would be wrong too, because probability cannot exceed 1. So it cannot be infinite either. But it can be *infinitesimal* (_very close_ to 0, but _not yet_ 0). Infinitesimal is not the same as infinite.
_"Look at it from a mathematical and not a physical science perspective"_
That's exactly the perspective I am looking at it.
_"it is just an infinite example"_
Infinite or not, it cannot and shouldn't lead to contradictions.
See also my objections to his next lecture. There's more of it.
+Bon Bon Infinity in itself is a paradox if treated as a real number. The area of that square is not defined therefore can be treated as infinite and a point that WAS defined would have a probability of 0 as the area of the square is infinite. A point by itself has no area as defined by geometry and therefore when testing for it's occurrence on an area must =0.
Alex Buck Infinity is a paradox only when treated as a _number_ (a number in general, not just a real number), which is incorrect. That's why it _shouldn't_ be treated as a number, and it is an error to do so. There is, however, nothing paradoxical in infinity itself. It is just an idea that tells you that you can continue some process without bounds. That's how all the ancient philosophers treated it: as a _potential_, not as something you actually need to walk through.
And the cause of the paradoxes in this probability problem is not because of the infinities (infinitely many points in a square), but because of treating points as if they had areas, which contradicts their definition: Points have no dimensions, so they don't have any area. (Have you seen my comment under his other lecture yet? I explain it there in more detail.)
_"The area of that square is not defined"_
What? :| What do you mean it's "not defined"? It's a 1x1 square, Celestia dammit! It's area is 1.
_"therefore can be treated as infinite"_
First of all: _non sequitur_. Second: no, it cannot be treated as infinite. It is 1, a finite number.
It is the _number of points_ on the square which is infinite, but this has nothing to do with areas at all.
_"A point by itself has no area as defined by geometry"_
Exactly.
_"and therefore when testing for it's occurrence on an area must =0."_
Not necessarily. Having no area doesn't necessarily mean area = 0. Thoughts have no area as well, but it doesn't mean that they have area of 0, because they are not geometrical objects either.
The problem with this approach lays not in infinities, but in the assumption that the square is _made of_ points (infinitely many), and therefore that the area of the entire square (1 in this case) is a sum of the areas of points (which are assumed to be 0). Too many incorrect assumptions, and it always leads to paradoxes.
The original assumption was that probabilities in that square are measured by areas (the area of the entire square being the primary example: equal to 1). So talking about the areas of the individual points on that square is nonsense, because it contradicts that assumption. We are supposed to measure probabilities with areas, not with points, right? We can subdivide the square into smaller and smaller areas, but they will _never_ be points. They will always be some smaller areas (of polygons or some other geometric figures). They can shrink around some point only _in a limit_, but this is not the same as _becoming_ a point (approaching is not the same as becoming). Therefore, infinitesimal areas of smaller and smaller shapes are OK, 0 areas of points are not OK.
There are no paradoxes. Learn something about calculus and you will see it's pretty exact and rigid science. In the context of this science, infinity exists, it's well defined, and what the professor said is true...
Thank you MIT ! , thank you John Tsitsiklis ! , very good and interesting lecture .
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.
The discussions are clear and concise. Appreciate it
Beautifully delivered lectures. The content is well structured and easy to review.
What's the probability of someone entering the wrong lecture theatre at 28:37 ?
It's great to have seen Richard Dawkins give a lecture about probability😀😃
Just started my probability journey at MIT OCW
Εξαιρετικη δουλεια Κε Τσιτσικλη.
Great Lectures!! Really nice
But Reading the same books for this along made it more comprehensible
Thanks MIT. for your great help
Great Work .. Really appreciate you making the knowledge available to the world.
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.
respect from hungary. wish i have you in my university instead of prof. szegedi gabor
Start at 10:40
16:53. The sequential v. matrix representation of the sample space looks like a game in extensive form and normal form.
Thankyou sir for very conceptual lecture thank you MIT open course feels like in the class awesome technical support
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.
If the size of the square is like the size of a football field, would you believe the space for a dart will be infinite then?
Eli Nope and according to what you say, what is the area of a point? It is the tip of a dart to be treated as a point, or not?
Regards.
"Real part of the lecture" starts at 9:55
Feichen Yang Thanks for pointing out.
His voice is so cool and relaxing
I love that he uses transparencies.
47:09, what would the sample space in terms of area look like for the coin flip of heads for the first time? How would you present an infinite area?
It is a discrete continuous process. There would be no area.
In these series, every topic is covered of this professor books written
Mister John our respect from Greece!! Pretty helpful courseware.. :)
Really interesting explanation, “You should not say sth if you don’t have to say it”
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 .....
+Sipho syphonic sthole Moyo Not a very easy school to get in to!!
+Jeffrey Stockdale Plus much cheaper just watching and learning on UTube!!
This man is a wonderful Guru
Absolutely an exceptionally perfect course! Thank you MIT!
Here after Priyansh's recommendation??
Here
Here
@@pablooctaviano4190 same
The tle one ?
appreciate for the open course from mit
at 33:40, why is it only true for union of finite number of sets?
Amazing lecture, such a great professor!
The captions at 32:15 are slightly off: instead of "manage", it should say "massage".
We've updated the caption. Thanks for the note!
22:30 that's pretty crazy that the probably of a specific point is 0, but the area of the sample space is > 0.
Take a look at the Mandelbrot set: it has a finite area, but an infinite length of the boundary
A new course...
THANKS MIT
:D
The real part of the lecture starts at 9:55
and the complex part?
@@ravichandramurthy461 😂
at 48:42, what algebra is he talking about in order to obtain 1/3 as a result of the sum of all probabilities?
Read up on geometric series.
probability is the framework for dealing with uncertainty or situation in which randomness occur.
countable additivity axiom.discrete uniform law.contiuous uniform law
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.
Thanks for sharing this video, it helped me a lot.
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
Very nice and concise explanation. Thanks.
1:42 i wonder what life-story the two pink-heads on the bottom left are cooking up..
What a superb Professor.
If I had only seen this during my Probability Course !!!!!!! at U of T
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
George E. you don't its stochastic as hell.
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.
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.
sorry for late answer, but if you still need it, it is 18.650 th-cam.com/video/VPZD_aij8H0/w-d-xo.html
i was just wondering if he changed the die experiment to say we have two colored dice, will this be considered as a similar example or experiment ?
Sir u said event a and event b should be independent then for subset 2,2 how can we use axiom principle
I was here in Sep 2024, 13 years after undergraduation.
skip till 10:00 to jump r8 into the lecture
i think at 44:58 it should be 1/4 + 1/4 = 1/2 not 1/2 +1/2 =1
+mohammad abdullah no you getting that wrong, it's a diagonal intersecting the x and y axis in (1/2,0) and (0,1/2)respectively, (.25,.25) is actually a point on that diagonal
+netfischer oh i was taking two points and adding there intercepts thanks for heip
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.
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.
Thanks for the videos! Any lecture on econometrics?
Thank you MIT
Also, the accent his so cool
Starts at 10:00
28:37 that was awkward. Looks like he actually left the class, poor guy.
11:25 19:03 21:03 24:42 36:00 40:34 44:44 48:58