Measure Theory 1 | Sigma Algebras

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  • Опубліковано 9 гру 2024

КОМЕНТАРІ • 359

  • @brightsideofmaths
    @brightsideofmaths  4 роки тому +47

    Please do the quiz to check if you have understood the topic in this video: thebrightsideofmathematics.com/measure_theory/overview/
    There is also a dark version of this video! ua-cam.com/video/xZ69KEg7ccU/v-deo.html

  • @lehoangsonsg7436
    @lehoangsonsg7436 4 роки тому +281

    What you are doing is amazing. I hope you can produce more content in English for non-German speakers.

    • @NegativeAccelerate
      @NegativeAccelerate Рік тому +12

      I honestly feel like learning German to get access to mroe videos

  • @roger9822
    @roger9822 5 років тому +80

    Amazing mini-course series, it helps a lot to get through probability theory. Although your videos are short and illustrative, you never lose mathematical rigidity. Thank you so much!

  • @Artonox
    @Artonox 4 роки тому +71

    THANK YOU GOD FOR BRINGING YOU INTO THIS WORLD.
    I finally found someone who can actually teach measure theory online! Ive always had this on my mind (my worst subject in mathematics, because i didnt understand my lecturer), and finally, nearly 8 years later, you made this beautiful video series for me to revisit and you explain very well.
    I did get a first class in the end, but I really was interested in measure theory and ashamed that i wasn't able to do this well. This serves as a second chance for me!

    • @NilodeRoock
      @NilodeRoock 2 роки тому +1

      Say things like "THANK YOU GOD FOR BRINGING YOU INTO THIS WORLD." to your parents, spouse or siblings, if you have to. Go and support the content provider financially if you want to say thanks. Just my 0,01c.

  • @zhaoyuzhu254
    @zhaoyuzhu254 3 роки тому +149

    I literately spent 12 mins on UA-cam and understand the whole thing, while I spent 2 hours on my professor's recording and still have no idea what he is talking about. :)

  • @perkelele
    @perkelele 4 роки тому +5

    Summary:
    A measure is a map of the generalized volume of the subsets of X.
    Power-set: set of all subsets of a set X. if X = {a,b} then P(X)={empty,X, {a}.{b}}
    Measurable Sets: We don't need to measure all the subsets we can form, only some of them. Can be the whole power-set, but is useful smaller. Useful because generalizing length in a meaningful way doesn't work for all sets, but only some sets.
    A is a Sigma Algebra: each element is a measurable set
    a) Empty set, and Full set are elements of A
    b) If a subset is measurable then so is its complement
    c) If every individual countable set is part of the sigma algebra, then the union of all these sets is also in the sigma Algebra
    To speak of an area of A we need for the sets that make it up to be measurable. So if you take all the individual sets (units) that make it up, you will get the whole.
    The smallest sigma algebra A = {emptyset, X} it validates all three rules.
    The largest sigma algebra A = P(X) because it contains all the subsets. In the best case scenario we can measure them all. But this is not the case so we are often between these two cases.

  • @mercymuenimwangi
    @mercymuenimwangi 3 роки тому +19

    It reached a point I just had to search measure theory for dummies. This is the best tutorial. I immediately subscribed and turned on notifications. Thank you so much

  • @cardinalityofaset4992
    @cardinalityofaset4992 2 роки тому +19

    Just a minor technical detail: You can slightly generalize the definition of sigma algebra by excluding the empty set from the first condition. Its presence in the sigma algebra immidiately follows from the fact that X must be measurable and that any complement of a measurable set is also measurable. (X^c = X \ X = 0 => 0 is measurable). Awesome list of vidoes, it´s intuitive and entertaining to watch :)

    • @johnnyq4260
      @johnnyq4260 Рік тому

      I bet when he wrote that he was thinking about topology.

  • @nreceda
    @nreceda 4 роки тому +7

    Just found your channel. I am taking a course this semester on Stochastic Processes and as far as I can tell, your explanations are much easier to understand so thank you thank you thank you thank you.

  • @gulshanamna3796
    @gulshanamna3796 2 роки тому +5

    You are the one who really can make students as well as teachers to understand measure theory in real meanings

  • @lebesgue-integral
    @lebesgue-integral Рік тому +3

    This is a nice video! I took measure theory in undergrad and I loved the subject, although it was so abstract. Your videos definetely will help this make make sense to many people!

  • @PostModernAlchemist
    @PostModernAlchemist 6 місяців тому

    I have no idea how youre making this subject so approachable for someone who took real analysis and abstract algebra 10+ years ago, but thank you! This is great!

  • @rmbennet
    @rmbennet 2 роки тому +1

    I came here a year and a half ago I couldn’t understand any of it after the first one or two videos. it’s remarkably more intuitive after abstract algebra and real analysis. It’s actually really interesting.

  • @gustafa2170
    @gustafa2170 4 роки тому +55

    I'll come back to this video when I'm stronger. Need more training.

  • @richardgreenhough
    @richardgreenhough Рік тому +1

    Tried to understand this from a book and didn't. This video enabled me to grasp this easily. Great video!

  • @jiahao2709
    @jiahao2709 3 роки тому +1

    You are the professor of MIT level, you video lectures should be accepted, respected, appreciated and advocated!!!!!!!!

  • @afrolichesmain777
    @afrolichesmain777 4 роки тому +3

    Wow great explanation for an introduction to sigma algebra. It’s my first time looking at this material. Looking forward to the rest of your videos on Measure theory!

  • @garrycotton7094
    @garrycotton7094 4 роки тому +8

    Great video :D. This reminds me of Group Theory in a way.
    Empty set, A in Fancy A is like the identity axiom.
    Complement of A in Fancy A is like the inverse axiom.
    Union of A_i in Fancy A is like the closure axiom.

    • @axelperezmachado5008
      @axelperezmachado5008 4 роки тому +1

      does this mean that a sigma-algebra is a group under union?

    • @deept3215
      @deept3215 4 роки тому +5

      @@axelperezmachado5008 It isn't because there is no inverse of union in the sigma algebra

    • @axelperezmachado5008
      @axelperezmachado5008 4 роки тому +1

      @@deept3215 True! Haven't realised

    • @Sciophile
      @Sciophile 4 роки тому +2

      @@axelperezmachado5008 It's an abelian group with respect to the operation of symmetric difference.

  • @Heisenberg8307
    @Heisenberg8307 2 роки тому

    The moment I realised this dude is giving a brief explanation on Measure Theory, I subscribed immediately.

  • @julianvillaquira4127
    @julianvillaquira4127 5 років тому +13

    Your channel is amazing! Thanks for the videos, they are very helpful to me since I will take a measure theory course next semester. New subscriber 😀.

  • @thedan2
    @thedan2 5 років тому +3

    Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!

  • @harshavajpayee7652
    @harshavajpayee7652 4 роки тому +7

    Now after watching this
    I can say that measure theory is measurable 😅
    Thanks for this wonderful video ❤️

  • @RenyxGhoul
    @RenyxGhoul 3 роки тому

    Even more succinct and concise than my lectures but I understand it a lot more. Wow. Thank you!

  • @sallanmega1
    @sallanmega1 2 роки тому

    This is actually a great explanation.
    Greetings from Spain!

  • @Thaizir
    @Thaizir 3 роки тому

    Thanks for taking the time to produce this content, it brought me back memories when I was studying this course.

  • @mongky9903
    @mongky9903 3 місяці тому

    I would like to make a example for better understanding for sigma algebra, correct me if I was wrong.
    Given X = {1, 2, 3} and sigma algebra A = {∅, X, {1}, {2, 3}}
    Let insert {2} to A to make A = {∅, X, {1}, {2, 3}, {2}}
    - But complement of {2} which is A \ {2} = {1, 3} ∉ A --> A not sigma algebra
    Let continue insert {1, 3} to A to make A = {∅, X, {1}, {2, 3}, {2}, {1, 3}}
    - But now a union between {1} and {2} is {1, 2} ∉ A --> A not sigma algebra
    Let add {1, 2} to make A = {∅, X, {1}, {2, 3}, {2}, {1, 3}, {1, 2}}
    - But complement of {1, 2} is {3} ∉ A --> A not sigma algebra
    Let add {3} into A making A sigma algebra again and also A is now the power set of X.

  • @DimiqBaba
    @DimiqBaba 8 місяців тому

    Thank you, straight after the lecture I watch your lectures.

  • @Cowux
    @Cowux 3 роки тому

    Best measure theory video on YT!!!!

  • @evionlast
    @evionlast 4 роки тому

    Clearly explained, basic examples, very good

  • @JJ-fb2lp
    @JJ-fb2lp 4 роки тому +67

    Jesus dude. I have not seen any one that can explain this topic better than you, which can mean two things. 1. They don't understand the topic 100% but is trying to teach someone. 2. They don't know how much to dumb it down for people who are just trying to understand this topic.

    • @NewCalculus
      @NewCalculus 4 роки тому +3

      You've never understood it because nonsense cannot be understood, only believed.

  • @shreykabir
    @shreykabir 2 роки тому

    Thanks a lot... I was interested in measure theory and wanted to learn more about it... This video has helped a lot making it easier for a high school to understand...

  • @Zubair622
    @Zubair622 7 місяців тому +1

    Extremely marvelous explanation really enjoyed it
    Please also upload lectures of complex analysis

  • @marekbalcerzak5255
    @marekbalcerzak5255 3 роки тому

    I love to watch your videos to get notion about the subject before reading handbook. Great job !

  • @Wynell
    @Wynell Рік тому

    Thank you so much here! I have an exam in a few days and you're literally saving me :)

  • @abhaykumar7626
    @abhaykumar7626 Місяць тому

    From India a great respect for you . Your videos are amazing

  • @watermelon-s6j
    @watermelon-s6j 4 роки тому +2

    This video helped me a lot, thank you!

  • @brittnihall1688
    @brittnihall1688 4 роки тому +1

    These videos are amazing and incredibly helpful!! Thank you SO much!!

  • @sejuprajapati2005
    @sejuprajapati2005 4 роки тому

    Thank you so much for all videos. Your teaching skill is amazing.

  • @giorgiozannini5626
    @giorgiozannini5626 5 років тому

    Thanks man you saved me! Studying Bayesian statistics now and couldn't wrap my head around the whole measure stuff. Thank you very much again!

  • @idontthinkso5966
    @idontthinkso5966 Рік тому

    My goodness, you really do have a video in everything I'm looking for.

  • @JuanRodriguez-tr6st
    @JuanRodriguez-tr6st 4 роки тому

    You are one of the best teachers I’ve had

    • @JuanRodriguez-tr6st
      @JuanRodriguez-tr6st 4 роки тому

      Came here for functional analysis, stayed for measure theory

  • @tropicalpajamas9986
    @tropicalpajamas9986 4 роки тому +3

    Very well explained with straightforward and intuitive examples.
    We neeeeeeeeed more of this exciting course in Mathematics.
    Keep it up~!!!

  • @КонстантинДемьянов-л2п

    My man, you kinda sound like Mimir from God of War. Loving it!

  • @nathanbarnard7896
    @nathanbarnard7896 3 роки тому

    Just starting measure-therotic probability theory and these are great :)

  • @davidwright8432
    @davidwright8432 4 роки тому +1

    Thanks; amazingly clear! I hope I'll be able to follow as easily when after a bit more development, you actually start do things with the ideas!

  • @Anteater23
    @Anteater23 4 роки тому +1

    Would you ever consider making a maths series on the subject of topology? Your videos are brilliant!

    • @brightsideofmaths
      @brightsideofmaths  4 роки тому +1

      Thanks! I want to do that, yes :)

    • @Anteater23
      @Anteater23 4 роки тому

      The Bright Side Of Mathematics :) topological spaces just seem harder to visualise than metric spaces for me. Metric spaces felt like a very natural concept.

    • @brightsideofmaths
      @brightsideofmaths  4 роки тому

      @@Anteater23 Topological spaces are also very natural. Often the concrete distances between points are not import but just the knowing which one is near or far.

  • @ssmhsasd
    @ssmhsasd Рік тому

    Very intuitive explanation, thank you. Very helpful for Engineers 👍

  • @mushroomsteve
    @mushroomsteve 4 роки тому +3

    Proposition: A sigma-algebra F is closed under finite intersections.
    Proof: Let F be a sigma algebra on a set X, and let A and B be elements of F. Then (A n B) = ((A^C u B^C)^C). Therefore, (A n B) is an element of F.
    Corollary: A sigma-algebra that is closed under arbitrary unions is a topology.

    • @sheerrmaan
      @sheerrmaan 4 роки тому +1

      Because the complements of A and B must belong to the sigma algebra by condition II. And the union of these two complements also belongs by III. And again the complement of it belong to F.

    • @mushroomsteve
      @mushroomsteve 4 роки тому

      @@sheerrmaan Exactly.

  • @ramkumarr1725
    @ramkumarr1725 3 роки тому

    Very nicely explained. And truly innovative to link it up to the quiz. I will try to see the other videos but the first chapter was very good.

    • @brightsideofmaths
      @brightsideofmaths  3 роки тому

      Thank you! I also want to do quizzes for the other parts if they are helpful.

    • @ramkumarr1725
      @ramkumarr1725 3 роки тому

      @@brightsideofmaths They are helpful for retention of material and application, IMHO. Math is not a spectator sport, IMHO.

  • @daviddavini847
    @daviddavini847 3 роки тому

    This was incredibly helpful, thanks for the knowledge!

  • @lexluthor6975
    @lexluthor6975 4 роки тому

    Very insightful explanations. Some people are born lucky!

  • @axelperezmachado5008
    @axelperezmachado5008 4 роки тому +1

    This channel is amazing! So glad i found it! I subscribed of course

  • @arefpourseyedi8087
    @arefpourseyedi8087 2 роки тому

    The presentation was amazing. Thank you!

  • @RAJ6118
    @RAJ6118 3 роки тому

    Beautiful insight of the topic

  • @user-ib4bg9kg5s
    @user-ib4bg9kg5s 3 роки тому

    The name is so scary, so we need people like you in this world to make them look less intimidating, thanks for the explanation

  • @YitzharVered
    @YitzharVered 3 роки тому

    I went through this crap in introduction to probability and was totally lost. Thank you for explaining.

  • @TDRT23
    @TDRT23 Рік тому

    THIS VIDEO IS AMAZING!!

  • @EebstertheGreat
    @EebstertheGreat 4 роки тому

    Part (a) of the definition is somewhat redundant with part (b). If we assume ∅ ∈ 𝒜 and that (A ∈ 𝒜) → (Aᶜ ∈ 𝒜), then ∅ᶜ = X ∈ 𝒜 by definition, so it is not required to include that in part (a).

    • @brightsideofmaths
      @brightsideofmaths  4 роки тому +2

      You are totally right! I often used redundancy is definitions to make them clearer.

  • @wesolyfoton
    @wesolyfoton 3 роки тому

    Great explanation. Kudos!

  • @_Navani_
    @_Navani_ 5 років тому +9

    This is such a helpful video! Now i feel like i can pass measure theory

    • @boyzrulethawld1
      @boyzrulethawld1 4 роки тому +2

      What course are u taking this for? 🤔

    • @cvdvdfhgh4946
      @cvdvdfhgh4946 4 роки тому

      @@boyzrulethawld1 id guess measure theory

  • @aoxinguo4256
    @aoxinguo4256 23 години тому

    thank you for sharing this amazing video! definitely love it

  • @pointofview6679
    @pointofview6679 4 роки тому

    Thanks so much sir.This is amazing and helpful

  • @jasonlim8387
    @jasonlim8387 2 роки тому

    Thank you, it was a very helpful video!

  • @syamalchattopadhyay2893
    @syamalchattopadhyay2893 4 роки тому

    Excellent video lecture

  • @IhrffhhgJgfhhbvf
    @IhrffhhgJgfhhbvf 6 місяців тому

    I finally understand why a sigma algebra is the way it is. The drawing made it so clear to me

  • @davidshechtman4746
    @davidshechtman4746 3 роки тому

    Answer me this. Cantors diagonal argument requires a square matrix to be certain that every entry is covered. The matrix (list, which ever. I use the word in a general sense) he proposes is based on permutative recombination. So for the universe of {a, b, c} I create a list of permutations abc, bca, cab, etc. Ordered in the manner of 3 columns and 6 rows. Iteration of one additional element, d, to the universe in consideration {a, b, c, d} will now produce a list with 4 columns and a page and a half of rows. The initial Alelph null of basic infinity we are guaranteed by Zermelo is won by Iteration. Clearly construction of a square matrix based on permutative recombination is impossible. How then, pray tell, does it magically occur for Cantor?

  • @최주희-n2w
    @최주희-n2w Місяць тому

    I love this lecture. Thank you :)

  • @Bolvarsdad
    @Bolvarsdad 8 місяців тому

    What's the difference between stating that the power set of X = {a,b} is P(x) = {{}, X, {a}, {b}}, and P(x) = {{}, {a}, {b}, {a,b}}? Reading Wikipedia, it told me that the latter notation is used, so I guess these are interchangeable?

    • @brightsideofmaths
      @brightsideofmaths  8 місяців тому

      There is no difference. Both sets P(X) from you are exactly the same.

  • @pk_1320
    @pk_1320 4 роки тому

    Loved it, great video!

  • @sebon11
    @sebon11 3 роки тому

    Great video, man!

  • @K4moo
    @K4moo 3 роки тому

    Thank you for the great video!! I think (c) is somewhat redundant because (b) says A and A^c (X\A) must both exist, therefore their union is X, and X is in the Sigma Algebra according to (a), the second veens diagram seems wrong in my understanding.

    • @brightsideofmaths
      @brightsideofmaths  3 роки тому

      Thank you! What exactly is wrong about the Venn diagram?

    • @K4moo
      @K4moo 3 роки тому

      @@brightsideofmaths I was thinking (c) is union of all elements in sigma algebra, which should be X. I rewatched this part, I think you actually meant the union of elements in one of the element in sigma algebra.

  • @hisxmark
    @hisxmark 4 роки тому

    The length is the absolute value of (b - a) = |(b - a)| = |(a - b)|. In other words, the length is a distance, a scalar greater than zero between any distinct points. So the length of (a - b) is equal to the length of (b - a).

  • @td_27
    @td_27 Рік тому

    Beni buraya kadar getiren eğitim sistemimize teşekkür ediyorum.

  • @varnita4455
    @varnita4455 3 роки тому +1

    Thank you.

  • @ludwig-music
    @ludwig-music 2 роки тому

    9:52 Are all sets of a sigma-algebra called "measurable sets", even the ones that are not measurable? "Proof": (a) the power set may include non-measurable sets, and (b) the power set is always a sigma-algebra; hence, (a) and (b) imply that it is possible that some sets of a sigma-algebra are not measurable, even though they are called measurable sets.
    Edit: I figured it out: If you already have a measure and some subsets that you can measure, then you should also be able to measure all the additional subsets that are required to make the family of subsets a sigma-algebra. So I think that's what the word "measurable" is referring to.

  • @monsijbiswal1104
    @monsijbiswal1104 4 роки тому

    Just awesome. Loved it!

  • @martinpuente7526
    @martinpuente7526 10 місяців тому

    amazing videos! I'm a econ student and I'm trying to deepen the subject. You said in 10:26 we need two elements of a subset to form a sigma-algebra. What if the subset is the empty set? that would be one element and it satisfies the three conditions

    • @brightsideofmaths
      @brightsideofmaths  10 місяців тому

      Thank you! I don't understand your question completely. Can you elaborate on that?

    • @Hold_it
      @Hold_it 10 місяців тому

      ​@@brightsideofmathsThe question basically boils down to: can the empty set be a Sigma Algebra?
      Meaning our Set X is just the empty set itself.
      In which case the Sigma Algebra would only consist of one element.
      The empty set.

    • @brightsideofmaths
      @brightsideofmaths  10 місяців тому +1

      Answer: Yes, it's possible but uninteresting ;)@@Hold_it

    • @Hold_it
      @Hold_it 10 місяців тому

      ​@@brightsideofmathsNice thank you❤
      The formulation that a sigma algebra needs at least two elements also made me unsure, but I get why it isn't really worth noting that this special case exists.

    • @martinpuente7526
      @martinpuente7526 10 місяців тому +1

      ​@@Hold_it yes! that was exactly what I meant thanks

  • @jatayubaxi4553
    @jatayubaxi4553 5 років тому

    Absolutely clear explanation.

  • @armatg
    @armatg 3 роки тому

    This is the first video where i understand what a sigma algebra is, thank you!

  • @rgoswami
    @rgoswami 4 роки тому

    Fantastic work!

  • @babumaths8066
    @babumaths8066 Рік тому

    Very useful. Thank you sir.

  • @abdulghanialmasri5550
    @abdulghanialmasri5550 3 роки тому

    Man, you are a great teacher 👍

  • @xPlosiveFobx
    @xPlosiveFobx 4 роки тому

    Thank you this helped so much!

  • @josmithephraim9078
    @josmithephraim9078 4 роки тому

    Wow!! These explanations are really nice. Immediately subscribed 👌

    • @josmithephraim9078
      @josmithephraim9078 4 роки тому

      What books can we refer to understand more about Measure theory, distribution functions, chebyshev lemme etc?

    • @brightsideofmaths
      @brightsideofmaths  4 роки тому

      There are a lot of books. I really like Schilling's about measures and other stuff :)

    • @josmithephraim9078
      @josmithephraim9078 4 роки тому

      @@brightsideofmaths Danke gut !

  • @MrOvipare
    @MrOvipare 3 роки тому

    I was reading Cybernetics : or control and communication in the animal and the machine by Norbert Wiener yesterday... I was surprised by the fact that I got totally lost when he used concepts of "measures" in the chapter about groups and statistical mechanics. What game is this!? What are those objects? Turns out I didn't knew shit about measure theory, despite my physics and engineering studies + working in R&D. This is exactly where I needed to land!

  • @pedromazariegos6536
    @pedromazariegos6536 Рік тому

    I love your work man

  • @rivology8423
    @rivology8423 3 роки тому

    Love you’re videos

  • @epsilondeltaclasses9126
    @epsilondeltaclasses9126 4 роки тому

    Best lectures from India

  • @tree3868
    @tree3868 2 роки тому

    Could you make more videos about the hysteresis system that described by measure theory

    • @brightsideofmaths
      @brightsideofmaths  2 роки тому

      Sounds like a very good idea. Do you have more details there what you want?

  • @rafaelb.333
    @rafaelb.333 4 роки тому +1

    So glad I've found your channel. Which book did you use to study this?

  • @psytno
    @psytno 4 роки тому

    thank you really I enjoy these topics

  • @KJ-ii1hq
    @KJ-ii1hq Рік тому

    Thank you! You save my life😭😭😭

  • @gym5959
    @gym5959 2 роки тому

    at 08:15 can you explain what do you mean by countable union of infinitely many sets ?

    • @brightsideofmaths
      @brightsideofmaths  2 роки тому +1

      The index set for the union is given by the natural numbers :)

  • @yujinlee8188
    @yujinlee8188 3 роки тому

    This is amazing!! Thank you so much 🙏

  • @shraddha5588
    @shraddha5588 3 роки тому

    Thank you ! its such a good explanation.

  • @ChloeYang-z3g
    @ChloeYang-z3g 3 місяці тому

    Amazing video! A question: for the plot you drew, should it be P(X) instead of X? As X is a set and P(X) is the set of all subsets, so A should be a part of all subsets.

    • @brightsideofmaths
      @brightsideofmaths  3 місяці тому

      A is a part of X and an element of P(X). So these are the same pictures but with different visualizations.

  • @gym5959
    @gym5959 2 роки тому

    at 7:19 in the venn diagram should it not be P(X) instead of X as A belongs to the A(italic) which has elements from the power set of X ?

    • @brightsideofmaths
      @brightsideofmaths  2 роки тому

      A is an element of P(X), but a subset of X.

    • @gym5959
      @gym5959 2 роки тому

      @@brightsideofmaths thank you so much , you are amazing ;)

  • @mobilekaki8402
    @mobilekaki8402 5 років тому +6

    Thank you so much. now finally i can understand it.

  • @JTan-fq6vy
    @JTan-fq6vy 9 місяців тому

    Thanks for the great video! Regrading the definition of sigma-algebra, why do we require the set X is measurable? Would it be possible that that a set X is *not* measurable while the its subsets are measurable? For example, we may not know exactly the volume of the universe but the space of the British Museum is somehow measurable. Any explanation or hint is really appreciated!

  • @tsunningwah3471
    @tsunningwah3471 Рік тому

    I am no math student and once tried to explain to my friend why it is impossible to pick a rational number on the real number line and here is my explanation. Imagine you are to create a number 0.xxxxxxxxx…by determining its decimal places at random, say drawing a number from 0-9. For example if you draw 3 ,2 and 6, then your number will be 0.326. Since there are infinitely many decimal places, the process goes on forever and you will likely be getting an irrational number. To create a rational number say 1/3 or 0.333333 that means you have to keep picking the same number forever which is impossible if you are picking the numbers at random. Is that a correct and good explanation?

    • @brightsideofmaths
      @brightsideofmaths  Рік тому

      Probability = 0 does not mean "impossible". I have a whole video series about that :)
      tbsom.de/s/pt