🎯 Key Takeaways for quick navigation: 00:00 📝 *Today's agenda includes a quick review, introducing conditional probability, and exploring its applications in breaking down problems and inference.* 07:11 🤔 *The misconception that arises when applying the additivity axiom to infinite sets is clarified, highlighting the distinction between countable and uncountable sets.* 09:31 💡 *Zero probability does not equate to impossibility; rather, it signifies extreme unlikelihood, as demonstrated in continuous models where zero probability outcomes are observed.* 14:40 🔍 *Conditional probability revises beliefs about outcomes based on new information, illustrated through numerical examples and a heuristic approach.* 16:33 📊 *The definition of conditional probability is expressed, linking it to the total probability of the conditioning event and the joint probability of both events.* 18:55 🔄 *Conditional probabilities adhere to the same principles as ordinary probabilities, ensuring that the revised probability model remains consistent and satisfies properties like additivity.* 20:52 🎲 *Conditional probabilities are applied to scenarios like two rolls of a die, where they facilitate the calculation of probabilities within a modified universe based on given conditions.* 24:13 🛫 *Conditional probabilities are essential in real-world situations, such as determining the likelihood of an airplane's presence based on radar data.* 27:31 📊 *Conditional probabilities enable the calculation of complex probabilities by breaking them down into simpler conditional events.* 29:46 🛰️ *Bayes' rule provides a systematic way to update beliefs or probabilities based on new evidence, essential for inference and learning from experience.* Made with HARPA AI
The professor artfully covers the fundamental idea of probability with deceptive ease. Many teachers easily confuse the students by writing the total probability theorem and Bayes' Theorem formulas first and try to fit in an example into the formula to explain the topic. One would lose intuition as to what is happening underneath those equation quickly and simply churn out the results by putting in values into the equation. But, the professor here only at the end mentions the name of Bayes and hardly uses the word Total probability theorem. He arrives at those generalisations with pure logic by intuitively applying multiplication rule, total probability theorem and Bayes’ rule to solve a problem at hand.
no you cant. a point is 0. a point is infinite. its an idea... The probability model is based on ideas that we only tailor to the real world using graphs and numbers, they're not the real world. So, a point in a mathematical theory is infinitesimally small, smaller than a quark or a string. How does a point do this? Because thats the law that is entailed by a mathematic concept. bottom line: the point you talk about is a point in the blackboard. its not the same point the professor is talking about.
I watched Probability for Life Science, Lecture 1, Math 3C, UCLA, it's a very good class as well. Those of you who loves Prob and Stats, you should watch too
I think we can assume the probability of individual points in continuous sample space as epsilon which is infinitely small but when add it turns out to be 1 without sacrificing the math/probability
rama chandra sadly that can't work, for two reasons The first is that ε can be infinitely small but isn't a single value, instead it is used to represent different radii that can potentially get infinitely small Second, what you're saying is defining something infinitely small multiplied with something infinitely big to be equal to 1 which doesn't behave nicely with other established things in maths It is however a very interesting thought experiment
No, you cannot. It REALLY is zero. If you give the uncountably many any value, they'll add up to more than 1. Before the experiment, every real number really has a chance of 0 to occur, only afterwards one will have occured.
At approximately 41:30 why are the two bottom branches not filled out like the first? Is there a reason A_2 and A_3 do. It have an intersection with B^c or is it supposed to be implied?
p(a and b) = p(a)*p(b|a). The probability that both a and b occur, when only one event can occur is: given that a occurs the probability that BOTH a and b occurs is the probability of a occuring multiplied by the probability that b has also occured given that a has occured. On the other hand, if it given that B has occured, then the probability that both a and b have occured is the probability that b occured multiplied by the probability and a has also occured given that b has occured.
Great lecture! 13:22 and then on paper: "P(A | B) undefined if P(B) = 0" But what if B contains two points and A contains one of them? P(B) = P(A) = 0, but I would like to say that P(A | B) = 1/2 What do you think about it?
Not sure you understand what is being said here. P(B) = 0 means that there are no points in B. The assumption so far has been that all points are equally likely so your example is not possible. P(A | B ) means the probability of A given that B occurred. If P(B) = 0 means B is impossible which would mean P( A | B ) makes no sense.
Prof Tsiksiklis said that an event with a probability of zero doesn't mean it is impossible to happen. It's just extremely unlikely to happen, almost impossible to happen, I'd say. Almost.
Игорь Коняхин how can B contain an event and still be zero probability unless, by point you mean in reference to the cartesian coordinates (x,y) in the continuous case..... in that case your entire sample space has to be continuous, thus there no countable points.
+Tinos Madzivire What i think he means is that either X or Y can be min 2. So that subspace would be like (2,2), (2,3), (2,4) along with (3,2), (4,2) - where either the value of 'X' or 'Y' is 2. And thats what he has highlighted as B in that figure. (If you look at the co-ordinates of each of the shaded areas i think it will seem clearer!).
I am confused with his proof that 1 = 0 and the reason why it's incorrect. He says that this is because the set of real points is uncountable (i.e. not a sequence). But what if we take a discrete grid of points (infinitely many), or for simplicity a 1D line of discrete points, i.e. the set of all integers Z. Then again we have that P(i) = 0, because there are infinitely many of them and they are all equally probable. But now, the set is countable, i.e. it is a sequence, and hence 1 = P(Ω) = sum(P(i)) = 0. What is wrong with this reasoning?
If a set is countable, then it must mean it is discrete and therefore the underlying distribution is a PMF and not a PDF. This also means that for each element in that countable set, there is no infinitesimal probabilities paradox. This is the reason why the additive property works in countable sets and not in uncountable sets.
Your reasoning about the square is flawed as by dividing it into a grid you turned it into a countable set. The sum of the individual probabilities is no longer 0 as it is now possible to assign finite probabilities for each cell in the grid. It only worked on uncountable sets, since it is not possible to assign independent probabilities without utilizing an infinitesimal and the fact that the infinity is much larger. This is a very subtle topic, so ask me questions to clarify. I am just an amateur, so I don't know much.
After some search, I think it has to do with the ‘weird sets’ and I think they are the null sets; and since the rationals are a null set, then they are part of the null set. (There’s a video I liked about null sets with the Banach-Tarski paradox)
i have a question my interviewer asked this but i don't know the answer "number of planes in the air right now how can we calculate this " can anyone guide me through this question
Hey you guys, I am looking for someone who has the book and check a specific example please on page 28 example 1.14. Probability of Alice being up to date if she was behind the previous week is given as 0.6 but writing the equations they have written P(U3|B2) as 0.4 . Is it a mistake??? It must be? I am confused. If someone can look into it, would be much appreciated. Thank you!!
I have taken several Probability and Random Proceses at different stages in my academic background career as well as many Advanced Stochastic Processes in many Electrical Engineering fields [Control Systems, DSP; Communication Systems, Radars, Electromagnetics, Machine Learning, DSP....]. It is really a pleasure to attend virtually to the lectures of Professor John Tsitsiklis. He makes probability a wonderful experience !!! Pd. I have a question for him that makes me think: Normally advanced sophisticated radar systems have PROBABILITY OF FALSE ALARM in the order of 10^-6 to 10^-10. Though his probability decision tree for the DETECTION of a plane is a wonderful really great and pretty neat example for the derivation of conditional probabilities, why did You use only a probability of false alarm [PFA] of 0.10? Isn´t it too high for advanced state of the art Radar Systems receivers Rx?
Any chance someone could help me out with this? I posted this on a separate MIT conditional probability video where scenario was: "1 blue marble 4 red in Cup A vs 3 blue 2 red in Cup B" and conditional probability of picking from cup A given there is blue marbe was said to be 1/4" because you add all of the blue marbles for denominator and add up those blue marbles in Cup A for the numerator. But I'm wondering if let's say I have 1 and only 1 marble in cup A, and it's a blue marble, and say 5 out of 11 marbles in cup B are blue marbles. It feels like if I know I picked a blue marble, then there should be more than a 1/6 chance of that marble coming from cup A. I guess because since cup A i this case is 100 percent blue? I don't know...
Interesting proof: showing probability of two separate coincidental events being larger than the probability of a single event occurring in the universe of events by saying that they are not coincidental events, however, but 1 single event with a larger probability than a single event occurring in the specific universe. Very clever.
min(X,Y) compares the two numbers X and Y and gives you the smaller one (or the minimum), so for example min(2,3) = 2 because 2 is smaller than 3. Similarly max(X,Y) compares the two numbers X and Y and outputs the bigger one (or the maximum), so for example max(2,3) = 3.
The meat begins at 12:00. 05:30 Errr, no, the square is not "made of" points. It *contains* those points. A point just marks a certain location inside that (pre-existing) square. One _cannot_ build the area of the square from points, because POINTS DON'T HAVE ANY AREA BY THEMSELVES!!! Areas could be made only from smaller (e.g. unit) areas. Otherwise there would be no difference between a scalar, a vector, a pseudovector, and a pseudoscalar in Clifford Algebras etc. There's a reason we distinguish those dimensions, right? :P The ridiculousness of that statement comes out nicely at 06:10 where he writes the sum of the probabilities for these points. He said in the previous lecture that the probability of a point is 0 (because it has no area, and we assumed that area = probability in this model), and he restated this assumption again in this lecture here. So we're actually *adding up 0s here*! And we _cannot_ get a positive result (in this case: 1) from summing up zeros, no matter how much of them we add up (be it 10, 100 or infinitely many). So the right side of this equation cannot be equal to the left side, and the entire equation is false. 06:40 No, the mistake is not in this point. The union is fine, as long as we're adding up actual areas. The mistake is in assuming that a point can have an area :P 07:20 Of course I can! Cantor's diagonalization argument is one example. Another one is Hilbert's space-filling curve. Long story short: yes, there are ways of ordering points on a surface, as well as in space, by threading them on a line. In other words, there exist functions which map points on a surface (or in space) onto a line. And this is another proof that this is not what is wrong with this argument. The "points have area" assumption is. 17:20 If the probability of B is 0, then we're contradicting ourselves, because the "|B" in "P(A|B)" was supposed to mean that B occurred *for sure*. So it _cannot_ be 0! The very statement that "B occurred" means that this probability is _not_ 0, and we are OK to put it in the denominator. Logic, Mr. Titskills :P
For 7:20 I think what he means is that you can find a way to arrange points on that surface, but the number of points needed to cover the whole surface is more than countable infinity. This tells us that we are dealing with a set whose cardinality is greater than aleph naught, which proves that probability cannot be applied to the problem. At least not with the current material covered. P.S: I am not saying you're wrong or anything, just trying to make sense of your explanation and his.
sureca18 Why trying to make sense of something which is nonsense? Just because some university professor says so doesn't make it less absurd. I explained it in more detail in several comments under the next video because he repeats the same error there and someone asked me about that. It is even more evident there, because he explicitly defined probability as areas, and then tried to apply his logic to points which obviously don't have area (it's basic geometry; the Definitions in the first pages of Euclid's "Elements", to be exact). Using areas inside that square to represent probabilities of hitting that area is fine, and these areas can be as small as we want, but however small they be, they will never be points. So counting points is irrelevant here, because points violate his definition of probability for that problem, and since they don't have areas and they are not areas themselves, using them would contradict this definition anyway. It's pretty much the same problem as with those people who think that lines are "made up of points" and try to calculate lengths of line segments by trying to count the number of points on them. It simply cannot work, because one can have two line segments of different lengths, and still one can make a 1:1 map between all the points from one segment and all the points from the other, so the number of points on both is the same (uncountably infinite), and the number of points is irrelevant to their lengths. One can even come up with a 1:1 map of all the points in a plane and all the points on a line. Lengths (as well as areas & volumes) come from something else entirely: from comparing them with units of length (or area or volume etc.).
abhineet sharma Yes, I've read it. THOROUGHLY. And it's hard _not_ to be frustrated seeing how university professors are ignorant of elementary geometry.
0 probability can mean impossible because you will never be able to measure with infinite precision, so you never can tell for sure that you hit a certain point.
![Putnam Product in an Integral??? [MIT Integration Bee 2015]](http://i.ytimg.com/vi/itLQTzTCedk/mqdefault.jpg)
This single lecture is like the backbone of this subject.
Great revisioh
This MIT probability video playlist changed my way of thinking towards probability.... ✌✌✌
Thank you Dr. Tsitsiklis. This lecture is a masterpiece.
This is the only lecture you will ever need to understand conditional probability
🎯 Key Takeaways for quick navigation:
00:00 📝 *Today's agenda includes a quick review, introducing conditional probability, and exploring its applications in breaking down problems and inference.*
07:11 🤔 *The misconception that arises when applying the additivity axiom to infinite sets is clarified, highlighting the distinction between countable and uncountable sets.*
09:31 💡 *Zero probability does not equate to impossibility; rather, it signifies extreme unlikelihood, as demonstrated in continuous models where zero probability outcomes are observed.*
14:40 🔍 *Conditional probability revises beliefs about outcomes based on new information, illustrated through numerical examples and a heuristic approach.*
16:33 📊 *The definition of conditional probability is expressed, linking it to the total probability of the conditioning event and the joint probability of both events.*
18:55 🔄 *Conditional probabilities adhere to the same principles as ordinary probabilities, ensuring that the revised probability model remains consistent and satisfies properties like additivity.*
20:52 🎲 *Conditional probabilities are applied to scenarios like two rolls of a die, where they facilitate the calculation of probabilities within a modified universe based on given conditions.*
24:13 🛫 *Conditional probabilities are essential in real-world situations, such as determining the likelihood of an airplane's presence based on radar data.*
27:31 📊 *Conditional probabilities enable the calculation of complex probabilities by breaking them down into simpler conditional events.*
29:46 🛰️ *Bayes' rule provides a systematic way to update beliefs or probabilities based on new evidence, essential for inference and learning from experience.*
Made with HARPA AI
The professor artfully covers the fundamental idea of probability with deceptive ease. Many teachers easily confuse the students by writing the total probability theorem and Bayes' Theorem formulas first and try to fit in an example into the formula to explain the topic. One would lose intuition as to what is happening underneath those equation quickly and simply churn out the results by putting in values into the equation. But, the professor here only at the end mentions the name of Bayes and hardly uses the word Total probability theorem. He arrives at those generalisations with pure logic by intuitively applying multiplication rule, total probability theorem and Bayes’ rule to solve a problem at hand.
Using the radar example instead of talking in "math speak" helps so much!!
This is the best MIT lecture on probability
Genius. I am so happy this is on the internet.
Thank you professor Tsitsiklis, It is clear and simple.
This is the best lecture I have ever listed to. Thank you Professor.
Thank you so much for such helpful content! I really appreciate being able to self-learn through these videos.
One of the Best lecture for understanding Conditional Probability and Bayes theorem.
Great example and lecture pace. Thank you Dr. Tsitsiklis and *****
Great professor, makes these concepts so much easier to understand
Multiplication rule/ divide and conquer/ Bayes' Rule
i think this lecture changed my life ...i now see education in a morer good perspection...by the way am from south africa ..stellenbosch university
at 15:59 he says probabilities should add up to zero. I think he meant one?
Yess
At 44:40 the girlfriend broke up with the guy with the cell phone or the class was too hard to understand?
Probably both. Life is hard.
This is MIT lecture, on probability.
he is writing notes
Observe him at 19:00
At 46:28 he is still mourning for his gf :)
P(B/A) would be Sensitivity in Epidemiology. P(Bc/Ac) would be Specificity. This is a great lecture.
...one of the best lectures I ever had.
...and I've out of school for tens years.
So elegantly explained. You are a master, Professor Tsitsiklis
Best discussion on Baye's, I have come across.
Super clear explanation of conditional probability and Bayes' theorem. Loved every minute of this video.
Thank you MIT and thank you professor John Tsitsiklis
Great Lecture, boiled down to simplicity :) Thank you prof.
John Tsitsiklis is so great teacher!
The essence lies in the last 6 to 7 minutes. Now I understood schematically hows and whys of Bayes's rule
I love the sarcasm of his voice in 27:50 "In case you care" xd
I saw a black lady in the course.
No obscure boring definition but still looks very clear with live examples, thank you!
This man knows to make discussion about probability fun!
thanks a lot John Tsitsiklis
, you are great professor
I literally screamed why is this prof. so **** smart..
The probability that 0 will happen is 1. It is absolutely likely to happen that you will get zero when we sum an infinite set of zeros.
no you cant. a point is 0. a point is infinite.
its an idea... The probability model is based on ideas that we only tailor to the real world using graphs and numbers, they're not the real world. So, a point in a mathematical theory is infinitesimally small, smaller than a quark or a string. How does a point do this? Because thats the law that is entailed by a mathematic concept.
bottom line: the point you talk about is a point in the blackboard. its not the same point the professor is talking about.
This man saved my GPA.
I watched Probability for Life Science, Lecture 1, Math 3C, UCLA, it's a very good class as well. Those of you who loves Prob and Stats, you should watch too
P({x,y | 0
I think we can assume the probability of individual points in continuous sample space as epsilon which is infinitely small but when add it turns out to be 1 without sacrificing the math/probability
rama chandra sadly that can't work, for two reasons
The first is that ε can be infinitely small but isn't a single value, instead it is used to represent different radii that can potentially get infinitely small
Second, what you're saying is defining something infinitely small multiplied with something infinitely big to be equal to 1 which doesn't behave nicely with other established things in maths
It is however a very interesting thought experiment
Actually I think that is exactly the idea of infinitesimal calculus ;) en.wikipedia.org/wiki/Hyperreal_number
No, you cannot.
It REALLY is zero. If you give the uncountably many any value, they'll add up to more than 1.
Before the experiment, every real number really has a chance of 0 to occur, only afterwards one will have occured.
At approximately 41:30 why are the two bottom branches not filled out like the first? Is there a reason A_2 and A_3 do. It have an intersection with B^c or is it supposed to be implied?
it's implied
Prof is great. MIT is very good.
p(a and b) = p(a)*p(b|a).
The probability that both a and b occur, when only one event can occur is:
given that a occurs the probability that BOTH a and b occurs is the probability of a occuring multiplied by the probability that b has also occured given that a has occured.
On the other hand, if it given that B has occured, then the probability that both a and b have occured is the probability that b occured multiplied by the probability and a has also occured given that b has occured.
33:35 three rules
Great lecture!
13:22 and then on paper: "P(A | B) undefined if P(B) = 0"
But what if B contains two points and A contains one of them?
P(B) = P(A) = 0, but I would like to say that P(A | B) = 1/2
What do you think about it?
Not sure you understand what is being said here. P(B) = 0 means that there are no points in B. The assumption so far has been that all points are equally likely so your example is not possible.
P(A | B ) means the probability of A given that B occurred. If P(B) = 0 means B is impossible which would mean P( A | B ) makes no sense.
gregg4 watch lecture one mate...
Prof Tsiksiklis said that an event with a probability of zero doesn't mean it is impossible to happen. It's just extremely unlikely to happen, almost impossible to happen, I'd say. Almost.
Игорь Коняхин how can B contain an event and still be zero probability unless, by point you mean in reference to the cartesian coordinates (x,y) in the continuous case..... in that case your entire sample space has to be continuous, thus there no countable points.
I did not understand at 30:15 he says there is a shortcut where look at the relative odds of these two events. how would this 'shortcut' work?
Well, he pretty much did it, the shortcut was you didnt have to think about it. you just use the formula. its the same solution.
Tested corona positive= 𝗣𝗔𝗡𝗜𝗞
After 33:15 = 𝗞𝗔𝗟𝗠
12:18 Conditional Probability
at 21:04 , the min(X,Y)=2, does the red shaded area cover the whole area of event B? can someone explain, I'm confused
+Tinos Madzivire What i think he means is that either X or Y can be min 2. So that subspace would be like (2,2), (2,3), (2,4) along with (3,2), (4,2) - where either the value of 'X' or 'Y' is 2. And thats what he has highlighted as B in that figure. (If you look at the co-ordinates of each of the shaded areas i think it will seem clearer!).
there is always someone who is unhappy as he said!
3:50 "Useful to have in our hands" ...But where is his right hand ???? o.O
I am confused with his proof that 1 = 0 and the reason why it's incorrect. He says that this is because the set of real points is uncountable (i.e. not a sequence).
But what if we take a discrete grid of points (infinitely many), or for simplicity a 1D line of discrete points, i.e. the set of all integers Z. Then again we have that P(i) = 0, because there are infinitely many of them and they are all equally probable. But now, the set is countable, i.e. it is a sequence, and hence 1 = P(Ω) = sum(P(i)) = 0. What is wrong with this reasoning?
If a set is countable, then it must mean it is discrete and therefore the underlying distribution is a PMF and not a PDF. This also means that for each element in that countable set, there is no infinitesimal probabilities paradox. This is the reason why the additive property works in countable sets and not in uncountable sets.
Your reasoning about the square is flawed as by dividing it into a grid you turned it into a countable set. The sum of the individual probabilities is no longer 0 as it is now possible to assign finite probabilities for each cell in the grid. It only worked on uncountable sets, since it is not possible to assign independent probabilities without utilizing an infinitesimal and the fact that the infinity is much larger. This is a very subtle topic, so ask me questions to clarify. I am just an amateur, so I don't know much.
Thank you, this is mindblowing
This Lecture is amazing!
(Completed)
amazing explanations
"always expect the unexpected" 😅 This prof rocks!!!
Teacher, you are the king!thank you!
Very clear lecture. Awesome job.
What if we consider only rational numbers inside the square? In this case, the set is countable but individual probabilities are still zeroes.
Have you found the answer? I’m also interested in it
After some search, I think it has to do with the ‘weird sets’ and I think they are the null sets; and since the rationals are a null set, then they are part of the null set. (There’s a video I liked about null sets with the Banach-Tarski paradox)
i have a question my interviewer asked this but i don't know the answer
"number of planes in the air right now how can we calculate this "
can anyone guide me through this question
Marvellous explanations
Why are MIT students having class in a David Lynch movie??
Hey you guys, I am looking for someone who has the book and check a specific example please on page 28 example 1.14.
Probability of Alice being up to date if she was behind the previous week is given as 0.6 but writing the equations they have written P(U3|B2) as 0.4 . Is it a mistake??? It must be? I am confused. If someone can look into it, would be much appreciated. Thank you!!
What is the reason for wearing a baseball cap backwards?
In bayes theorm if I know the value of p(symptoms/disease) let's 0.3 so,can I take p(~ymtm/dise)= p(ymtm/~dise) = (1-p(symtm/dis))?
Event B is the whole shaded part, right?
Not just the part under A3?
44:43 Bayes' rule
best explain i have seen yet !!!!
I have taken several Probability and Random Proceses at different stages in my academic background career as well as many Advanced Stochastic Processes in many Electrical Engineering fields [Control Systems, DSP; Communication Systems, Radars, Electromagnetics, Machine Learning, DSP....]. It is really a pleasure to attend virtually to the lectures of Professor John Tsitsiklis. He makes probability a wonderful experience !!! Pd. I have a question for him that makes me think: Normally advanced sophisticated radar systems have PROBABILITY OF FALSE ALARM in the order of 10^-6 to 10^-10. Though his probability decision tree for the DETECTION of a plane is a wonderful really great and pretty neat example for the derivation of conditional probabilities, why did You use only a probability of false alarm [PFA] of 0.10? Isn´t it too high for advanced state of the art Radar Systems receivers Rx?
so much clarity
Fantastic Lecturer, very clear..
what if you did not check the plane in the sky?
Wow, super clear! Really enjoy the lecture
Any chance someone could help me out with this? I posted this on a separate MIT conditional probability video where scenario was: "1 blue marble 4 red in Cup A vs 3 blue 2 red in Cup B" and conditional probability of picking from cup A given there is blue marbe was said to be 1/4" because you add all of the blue marbles for denominator and add up those blue marbles in Cup A for the numerator.
But I'm wondering if let's say I have 1 and only 1 marble in cup A, and it's a blue marble, and say 5 out of 11 marbles in cup B are blue marbles. It feels like if I know I picked a blue marble, then there should be more than a 1/6 chance of that marble coming from cup A. I guess because since cup A i this case is 100 percent blue? I don't know...
What's the difference between Bayes' theorem and conditional probability?
Human Evolution it is somewhat extended version of conditional probability
It is reverse conditional probability
Interesting proof: showing probability of two separate coincidental events being larger than the probability of a single event occurring in the universe of events by saying that they are not coincidental events, however, but 1 single event with a larger probability than a single event occurring in the specific universe. Very clever.
Splendid.
What does it mean min(X,Y) and max(X,Y) in 22:00 ?
min(X,Y) compares the two numbers X and Y and gives you the smaller one (or the minimum), so for example min(2,3) = 2 because 2 is smaller than 3. Similarly max(X,Y) compares the two numbers X and Y and outputs the bigger one (or the maximum), so for example max(2,3) = 3.
Frederik Rentzsch Thanks 🌷
The lecture is amazing
Best explanation EVER.
Text book recommendations anyone ?
ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/readings/
Thanks for MIT OCW
First Solved Example: 20:42
He keeps mentioning multiplication rule of probability here , but where did he teach that?
Lecture 1 should help: th-cam.com/video/j9WZyLZCBzs/w-d-xo.html
Awesomely done prof. XD.
Excellent explaination
is it possible to post old tests, exams or assignments? t'would be much appreciated.
View the complete course on MIT OpenCourseWare for the full materials (Readings, Lecture Notes, Tutorials, Assignments, Exams): ocw.mit.edu/6-041F10
+MIT OpenCourseWare thank you so much
Perfect lecture 👌👌
The meat begins at 12:00.
05:30 Errr, no, the square is not "made of" points. It *contains* those points. A point just marks a certain location inside that (pre-existing) square. One _cannot_ build the area of the square from points, because POINTS DON'T HAVE ANY AREA BY THEMSELVES!!! Areas could be made only from smaller (e.g. unit) areas. Otherwise there would be no difference between a scalar, a vector, a pseudovector, and a pseudoscalar in Clifford Algebras etc. There's a reason we distinguish those dimensions, right? :P
The ridiculousness of that statement comes out nicely at 06:10 where he writes the sum of the probabilities for these points. He said in the previous lecture that the probability of a point is 0 (because it has no area, and we assumed that area = probability in this model), and he restated this assumption again in this lecture here. So we're actually *adding up 0s here*! And we _cannot_ get a positive result (in this case: 1) from summing up zeros, no matter how much of them we add up (be it 10, 100 or infinitely many). So the right side of this equation cannot be equal to the left side, and the entire equation is false.
06:40 No, the mistake is not in this point. The union is fine, as long as we're adding up actual areas. The mistake is in assuming that a point can have an area :P
07:20 Of course I can! Cantor's diagonalization argument is one example. Another one is Hilbert's space-filling curve. Long story short: yes, there are ways of ordering points on a surface, as well as in space, by threading them on a line. In other words, there exist functions which map points on a surface (or in space) onto a line. And this is another proof that this is not what is wrong with this argument. The "points have area" assumption is.
17:20 If the probability of B is 0, then we're contradicting ourselves, because the "|B" in "P(A|B)" was supposed to mean that B occurred *for sure*. So it _cannot_ be 0! The very statement that "B occurred" means that this probability is _not_ 0, and we are OK to put it in the denominator. Logic, Mr. Titskills :P
For 7:20 I think what he means is that you can find a way to arrange points on that surface, but the number of points needed to cover the whole surface is more than countable infinity. This tells us that we are dealing with a set whose cardinality is greater than aleph naught, which proves that probability cannot be applied to the problem. At least not with the current material covered.
P.S: I am not saying you're wrong or anything, just trying to make sense of your explanation and his.
sureca18 Why trying to make sense of something which is nonsense? Just because some university professor says so doesn't make it less absurd.
I explained it in more detail in several comments under the next video because he repeats the same error there and someone asked me about that. It is even more evident there, because he explicitly defined probability as areas, and then tried to apply his logic to points which obviously don't have area (it's basic geometry; the Definitions in the first pages of Euclid's "Elements", to be exact). Using areas inside that square to represent probabilities of hitting that area is fine, and these areas can be as small as we want, but however small they be, they will never be points.
So counting points is irrelevant here, because points violate his definition of probability for that problem, and since they don't have areas and they are not areas themselves, using them would contradict this definition anyway.
It's pretty much the same problem as with those people who think that lines are "made up of points" and try to calculate lengths of line segments by trying to count the number of points on them. It simply cannot work, because one can have two line segments of different lengths, and still one can make a 1:1 map between all the points from one segment and all the points from the other, so the number of points on both is the same (uncountably infinite), and the number of points is irrelevant to their lengths. One can even come up with a 1:1 map of all the points in a plane and all the points on a line. Lengths (as well as areas & volumes) come from something else entirely: from comparing them with units of length (or area or volume etc.).
sureca18
Have you read Euclid's Elements? Your silly frustration with the 'language' may be addressed by ghost of Euclid himself then.
abhineet sharma
Yes, I've read it. THOROUGHLY. And it's hard _not_ to be frustrated seeing how university professors are ignorant of elementary geometry.
35:52 that student was using phone in the class :)
You are great thanks mit
Thankyou professor
"Oh Kay"
Maza aa gaya bhai!
Amazing class!
12:10 the review is over.
Thank You
0 probability can mean impossible because you will never be able to measure with infinite precision, so you never can tell for sure that you hit a certain point.
very comprehensive
hwat?
Superb teacher!
airplane example was awesome ....